#6660771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660434
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1838 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:

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

Polycyclic presentation:

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

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, 6567967);