#6621032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(13,14,19,23)(15,24,27,35)(16,28,30,21)(17,22,18,33)(20,31,36,26)(25,40,43,39)(29,38,32,37)(34,42,44,41);(2,4)(6,10)(8,11)(13,15,17,31,19,27,18,26)(14,20,33,35,23,36,22,24)(16,25,32,41,30,43,29,42)(21,34,37,40,28,44,38,39);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,25)(17,32)(18,29)(19,30)(20,34)(22,38)(23,28)(24,39)(26,42)(27,43)(31,41)(33,37)(35,40)(36,44);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,23,33)(15,26,27,31)(16,29,30,32)(20,35,36,24)(21,37,28,38)(25,41,43,42)(34,39,44,40);(5,8,9,11)(6,7,10,12)(15,27)(16,30)(20,36)(21,28)(24,35)(26,31)(29,32)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528228);