#6671663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2174 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)(6,9)(7,11)(13,14)(15,23)(16,21)(17,20)(18,22)(19,26)(24,35)(25,41)(27,33)(28,44)(29,36)(30,39)(31,37)(32,40)(34,43)(38,42);(5,6,8,9)(7,10,11,12)(13,15,17,26)(14,19,20,23)(16,28,29,42)(18,24,31,27)(21,38,36,44)(22,33,37,35)(25,30,43,32)(34,39,41,40);(2,4)(15,24)(18,31)(19,33)(22,37)(23,35)(25,42)(26,27)(28,43)(30,32)(34,44)(38,41)(39,40);(5,7,8,11)(6,10,9,12)(14,20)(16,29)(19,23)(22,37)(25,43)(28,42)(30,32)(33,35);(6,9)(10,12)(13,16,17,29)(14,21,20,36)(15,25,26,43)(18,32,31,30)(19,34,23,41)(22,40,37,39)(24,42,27,28)(33,44,35,38)

Polycyclic presentation:

2^9|b^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6578859);