#6620001

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#24256	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(18,22)(19,25)(20,28)(23,29)(24,31)(26,30);(2,6)(7,8)(10,14)(11,12)(18,23)(22,29)(24,26)(30,31);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,18,21,23)(19,26,25,24)(20,29,28,22)(27,30,32,31);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(17,19,21,25)(18,24,23,26)(20,27,28,32)(22,30,29,31);(9,12,13,11)(10,16,14,15)(19,25)(24,26)(27,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6527197);