#6585028

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6492224);