#93256

order	512
nilpotency class	3
isoclinism	isoclinic class of #7333
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#322
commutator	C₈	C₈²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7014
prime radical	C₂³	#338
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

183 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=ci,e^a=ef,h^a=hi,d^b=df,g^b=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 452);