#6683733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2333 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,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, 6590929);