#6693721

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18,23,22)(19,24,29,28)(20,25,21,30)(26,31,32,27);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19,23,29)(18,24,22,28)(20,27,21,31)(25,26,30,32);(9,12,13,11)(10,14,15,16)(17,20,23,21)(18,25,22,30)(19,26,29,32)(24,31,28,27);(2,5)(6,8)(17,21,23,20)(18,25,22,30)(19,27,29,31)(24,32,28,26);(1,3,4,7)(2,6,5,8)(18,22)(19,28)(24,29)(25,30)(26,27)(31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600917);