#6909655

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

2119 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=fi,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, 6816851);