#6586011

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6493207);