#124099

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bd,d^a=df,d^b=dg,e^b=efg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31295);