#6608237

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=ehi

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, 6515433);