#93322

order	512
nilpotency class	4
isoclinism	isoclinic class of #93313
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1281
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄×#97	C₂²
socle	C₂	#7019
prime radical	C₂³	#338
LCS	G ▷ C₂×C₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,4,10,6,13,5,23,7,11,22,9,21,12,24,28)(3,14,16,8,19,25,31,27,20,29,17,26,18,32,15,30);(1,3,5,17,6,18,22,31,7,20,24,16,21,19,4,15)(2,8,9,14,12,27,23,25,11,26,10,29,13,30,28,32)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=g,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,d^b=dg,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 518);