C₂×#500352 #72308607

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#500352

Subgroups

	subgroup	G/H
center, socle	C₂²	#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#61260	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 66 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249754, C₂×#72307115
maximal normal subgroup of	C₂×#72249754, C₂×#72307115

GAP

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

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

C2×#500352 #72308607