#96469

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

666 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fgh,d^b=dh

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, 3665);