#6911441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fg,d^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, 6818637);