#6992894

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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

1810 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)(19,23)(20,25)(21,24)(22,26)(27,32)(28,31)(29,30);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,29)(21,28)(22,27)(24,31)(25,30)(26,32);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(18,24)(19,27)(20,21)(23,30)(25,26)(31,32);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(17,20,22,21)(18,25,26,24)(19,28,27,29)(23,31,32,30);(1,6,7,5)(2,3,8,4)(19,27)(23,32)(28,29)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900090);