#363089

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27,23,29,21,28)(19,26,20,25,31,32,30,24);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,21,30,23,31,22,20)(18,24,27,26,29,25,28,32);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,19,22,20,23,31,21,30)(18,25,28,26,29,24,27,32);(2,5)(6,8)(10,14)(13,15)(17,20,22,31,23,30,21,19)(18,26,28,24,29,32,27,25)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=eg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270285);