#6566305

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,29)(24,34)(25,30)(26,36)(27,31)(28,32)(33,35);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,24,27,33,26,32)(22,28,36,35,31,34,30,29);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,26,23,27,32,25,33)(22,29,30,34,31,35,36,28);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,22)(23,28)(24,35)(25,36)(26,30)(27,31)(29,32)(33,34);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,36)(23,24,33,32)(28,29,34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6473501);