#93501

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂³	#336
prime radical	C₂⁴	#97
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=h,e^2=g,h^2=i,b^a=bc,c^a=cf,e^a=egi,c^b=cg,d^b=dfi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 697);