#6912763

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1717 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819959);