#6690777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597973);