#6624781

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(17,18,23,19)(20,22,27,24)(21,25,29,26)(28,30,32,31);(2,6)(7,8)(10,14)(11,12)(17,19,22,27,23,18,24,20)(21,26,30,32,29,25,31,28);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,19,23,27,24,18)(21,28,30,26,29,32,31,25);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32);(9,12,13,11)(10,16,14,15)(21,29)(25,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531977);