#6915521

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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)(20,27)(21,23)(22,25)(24,28)(29,31)(30,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,25)(18,22,21,26)(20,27,29,31)(24,28,30,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,26,32,23,29,22,28)(18,24,25,31,21,30,19,27);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,21,23,18)(19,22,25,26)(20,24,29,30)(27,32,31,28);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,26)(18,25,21,19)(20,28,29,32)(24,31,30,27)

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=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, 6822717);