#6771012

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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)(9,10)(11,15)(12,14)(13,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(11,16)(13,15)(17,19)(18,22)(20,28)(21,26)(23,30)(24,32)(25,27)(29,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,23)(22,30)(24,31)(29,32);(2,5)(6,8)(18,23)(19,26)(24,31)(25,28)

Polycyclic presentation:

2^9|d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6678208);