#93996

order	512
nilpotency class	3
isoclinism	isoclinic class of #7333
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#322
commutator	C₈	C₈²
Frattini, cosocle	C₂×#346	C₂²
socle	C₂	#7137
prime radical	C₂³	#347
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,c^a=ci,e^a=eh,g^a=gi,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217187, #72217188, #72217203, …
maximal normal subgroup of	#72217187, #72217188, #72217203, …
maximal quotient of	C₂×#93996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1192);