C₂×#152886 #72321887

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#152886

Subgroups

	subgroup	G/H
center	C₂×C₁₆	#127
commutator	C₂²×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₁₆	C₂⁴
socle	C₂²	#13190
prime radical	C₂×#525	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 66 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72217181, C₂×#72224997, C₂×#72225005
maximal normal subgroup of	C₂×#72217181, C₂×#72224997, C₂×#72225005

GAP

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

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

C2×#152886 #72321887