#225230

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄²

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

658 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)(18,22,25,23)(24,28,31,30);(1,3,4,7)(2,5,6,8)(17,18)(19,23)(20,28)(21,25)(22,27)(24,26)(29,30)(31,32);(2,6)(5,8)(17,19,21,27)(18,23,25,22)(20,26,29,32)(24,30,31,28);(17,20,27,32,21,29,19,26)(18,24,23,30,25,31,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132426);