#6618825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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)(3,6)(4,5)(7,8)(17,18)(19,24)(20,29)(21,26)(22,31)(23,27)(25,30)(28,32);(2,5)(6,8)(10,14)(12,15)(17,19,23,20)(18,24,27,29)(21,30,22,28)(25,31,32,26);(9,10)(11,12)(13,14)(15,16)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,15,14,12);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fh,e^2=hi,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=eh

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, 6526021);