#6915423

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,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, 6822619);