#328804

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=e,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236000);