#6600727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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)(17,18,22,25)(19,26,27,32)(20,24,21,30)(23,28,31,29);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(19,27)(23,31)(26,32)(28,29);(10,13)(11,15)(17,19)(18,23)(20,29)(21,28)(22,27)(24,32)(25,31)(26,30);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32);(10,13)(11,15)(17,20,22,21)(18,24,25,30)(19,29,27,28)(23,32,31,26)

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6507923);