#6526681

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(23,31)(24,29)(25,30)(26,28)(27,36)(32,35)(33,34);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(22,28)(23,32)(29,33)(30,36);(7,13)(8,15)(10,16)(11,18)(21,23,27,29,26,32,25,33)(22,24,30,31,28,34,36,35);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24,26,34)(22,29,28,33)(23,30,32,36)(25,35,27,31);(7,13)(10,16)(14,19)(17,20)(21,25,26,27)(22,30,28,36)(23,33,32,29)(24,31,34,35)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6433877);