#353168

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₈	C₂³
commutator	C₂²	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=e,f^2=i,h^2=i,b^a=be,d^a=di

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260364);