#6914619

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 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)(17,18,21,26)(19,25,22,23)(20,28,30,32)(24,27,29,31);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,22)(18,23,26,25)(20,27,30,31)(24,32,29,28);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,31,21,30,19,27)(18,24,25,28,26,29,23,32);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,18)(19,23)(20,29)(21,26)(22,25)(24,30)(27,28)(31,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,19,21,22)(18,25,26,23)(20,27,30,31)(24,28,29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6821815);