#6566221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,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, 6473417);