#468804

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1259 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=gi,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376000);