#6678892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
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×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,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, 6586088);