#6874393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6781589);