#6828721

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,27,32)(20,28,29,25)(21,26,31,30);(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,21)(22,27)(23,32)(25,30)(26,28)(29,31);(2,5)(6,8)(10,14)(11,16)(17,20,23,28,22,29,18,25)(19,26,24,21,27,30,32,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,18,22,23)(19,24,27,32)(20,25,29,28)(21,30,31,26);(17,21,18,26,22,31,23,30)(19,25,32,29,27,28,24,20)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=fg,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

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, 6735917);