#6990186

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,29)(21,28)(22,26)(23,31)(24,30)(25,32);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,24,26,27)(20,28,31,25)(21,29,32,23);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,20,29,18,22,31,23,26)(19,28,32,24,30,25,21,27);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21)(18,25)(19,29)(20,24)(22,32)(23,30)(26,28)(27,31);(1,5,6,4)(2,7,3,8)(19,30)(21,32)(24,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6897382);