#225235

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132431);