C₂×2₋¹⁺⁸ #72258032

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁵

Aliases

direct product

C₂×2₋¹⁺⁸

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂⁸
commutator, Frattini, cosocle	C₂	C₂⁹
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂ ▷ C₁

8298013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 9 generators on 66 points:

(3,4,11,18)(5,19,24,12)(6,13,25,40)(7,14,21,34)(8,15,33,38)(9,16,35,46)(10,17,36,30)(20,47,49,37)(22,48,53,32)(23,31,45,39)(26,41,50,60)(27,42,54,61)(28,43,55,63)(29,44,56,57)(51,65,66,59)(52,58,64,62);(3,5,11,24)(4,12,18,19)(6,20,25,49)(7,8,21,33)(9,22,35,53)(10,23,36,45)(13,37,40,47)(14,38,34,15)(16,32,46,48)(17,39,30,31)(26,27,50,54)(28,51,55,66)(29,52,56,64)(41,61,60,42)(43,59,63,65)(44,62,57,58);(6,25)(13,40)(20,49)(26,50)(27,54)(28,55)(29,56)(37,47)(41,60)(42,61)(43,63)(44,57)(51,66)(52,64)(58,62)(59,65);(3,6)(4,13)(5,20)(7,26)(8,27)(9,28)(10,29)(11,25)(12,37)(14,41)(15,42)(16,43)(17,44)(18,40)(19,47)(21,50)(22,51)(23,52)(24,49)(30,57)(31,58)(32,59)(33,54)(34,60)(35,55)(36,56)(38,61)(39,62)(45,64)(46,63)(48,65)(53,66);(3,7)(4,14)(5,8)(6,26)(9,30)(10,16)(11,21)(12,38)(13,41)(15,19)(17,35)(18,34)(20,27)(22,31)(23,32)(24,33)(25,50)(28,57)(29,43)(36,46)(37,61)(39,53)(40,60)(42,47)(44,55)(45,48)(49,54)(51,58)(52,59)(56,63)(62,66)(64,65);(3,8)(4,15)(5,21)(6,27)(7,24)(9,31)(10,32)(11,33)(12,14)(13,42)(16,45)(17,22)(18,38)(19,34)(20,50)(23,46)(25,54)(26,49)(28,58)(29,59)(30,53)(35,39)(36,48)(37,41)(40,61)(43,64)(44,51)(47,60)(52,63)(55,62)(56,65)(57,66);(3,9)(4,16)(5,22)(6,28)(7,30)(8,31)(10,14)(11,35)(12,32)(13,43)(15,45)(17,21)(18,46)(19,48)(20,51)(23,38)(24,53)(25,55)(26,57)(27,58)(29,41)(33,39)(34,36)(37,59)(40,63)(42,64)(44,50)(47,65)(49,66)(52,61)(54,62)(56,60);(3,10)(4,17)(5,23)(6,29)(7,16)(8,32)(9,34)(11,36)(12,39)(13,44)(14,35)(15,22)(18,30)(19,31)(20,52)(21,46)(24,45)(25,56)(26,43)(27,59)(28,60)(33,48)(37,62)(38,53)(40,57)(41,55)(42,51)(47,58)(49,64)(50,63)(54,65)(61,66);(1,2)

Conjugacy classes & character table

514 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	2₋¹⁺¹⁰, C₂²×2₋¹⁺⁸, C₂×#72258201
maximal normal subgroup of	2₋¹⁺¹⁰, C₂²×2₋¹⁺⁸, C₂×#72258201
maximal quotient of	C₂²×2₋¹⁺⁸

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4,11,18)(5,19,24,12)(6,13,25,40)(7,14,21,34)(8,15,33,38)(9,16,35,46)(10,17,36,30)(20,47,49,37)(22,48,53,32)(23,31,45,39)(26,41,50,60)(27,42,54,61)(28,43,55,63)(29,44,56,57)(51,65,66,59)(52,58,64,62), (3,5,11,24)(4,12,18,19)(6,20,25,49)(7,8,21,33)(9,22,35,53)(10,23,36,45)(13,37,40,47)(14,38,34,15)(16,32,46,48)(17,39,30,31)(26,27,50,54)(28,51,55,66)(29,52,56,64)(41,61,60,42)(43,59,63,65)(44,62,57,58), (6,25)(13,40)(20,49)(26,50)(27,54)(28,55)(29,56)(37,47)(41,60)(42,61)(43,63)(44,57)(51,66)(52,64)(58,62)(59,65), (3,6)(4,13)(5,20)(7,26)(8,27)(9,28)(10,29)(11,25)(12,37)(14,41)(15,42)(16,43)(17,44)(18,40)(19,47)(21,50)(22,51)(23,52)(24,49)(30,57)(31,58)(32,59)(33,54)(34,60)(35,55)(36,56)(38,61)(39,62)(45,64)(46,63)(48,65)(53,66), (3,7)(4,14)(5,8)(6,26)(9,30)(10,16)(11,21)(12,38)(13,41)(15,19)(17,35)(18,34)(20,27)(22,31)(23,32)(24,33)(25,50)(28,57)(29,43)(36,46)(37,61)(39,53)(40,60)(42,47)(44,55)(45,48)(49,54)(51,58)(52,59)(56,63)(62,66)(64,65), (3,8)(4,15)(5,21)(6,27)(7,24)(9,31)(10,32)(11,33)(12,14)(13,42)(16,45)(17,22)(18,38)(19,34)(20,50)(23,46)(25,54)(26,49)(28,58)(29,59)(30,53)(35,39)(36,48)(37,41)(40,61)(43,64)(44,51)(47,60)(52,63)(55,62)(56,65)(57,66), (3,9)(4,16)(5,22)(6,28)(7,30)(8,31)(10,14)(11,35)(12,32)(13,43)(15,45)(17,21)(18,46)(19,48)(20,51)(23,38)(24,53)(25,55)(26,57)(27,58)(29,41)(33,39)(34,36)(37,59)(40,63)(42,64)(44,50)(47,65)(49,66)(52,61)(54,62)(56,60), (3,10)(4,17)(5,23)(6,29)(7,16)(8,32)(9,34)(11,36)(12,39)(13,44)(14,35)(15,22)(18,30)(19,31)(20,52)(21,46)(24,45)(25,56)(26,43)(27,59)(28,60)(33,48)(37,62)(38,53)(40,57)(41,55)(42,51)(47,58)(49,64)(50,63)(54,65)(61,66), (1,2));

C2×2−1+8 #72258032