#6541178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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

1701 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)(7,16)(8,13)(9,14)(10,12)(11,15)(17,20)(18,19)(21,22,27,30)(23,31,33,25)(24,32,29,26)(28,35,36,34);(5,7,11,10)(6,12,15,16)(8,18,9,17)(13,20,14,19)(21,23,32,35,27,33,26,34)(22,28,24,25,30,36,29,31);(2,4)(5,8,11,9)(6,13,15,14)(7,17,10,18)(12,19,16,20)(21,24,26,22,27,29,32,30)(23,25,34,28,33,31,35,36);(5,9,11,8)(6,14,15,13)(7,17,10,18)(12,19,16,20)(21,25,27,31)(22,23,30,33)(24,35,29,34)(26,28,32,36);(5,10,11,7)(6,12,15,16)(8,18,9,17)(13,19,14,20)(23,33)(25,31)(28,36)(34,35)

Polycyclic presentation:

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

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, 6448374);