#6500805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1981 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6408001);