#7149379

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1926 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)(7,11)(10,12)(13,14)(15,23)(16,20)(17,29)(18,21)(19,25)(22,30)(24,35)(26,36)(27,34)(28,41)(31,37)(32,39)(33,40)(38,43)(42,44);(6,9)(7,11)(13,15,17,25,18,26,30,39)(14,19,22,23,21,32,29,36)(16,24,28,37,27,38,40,44)(20,31,33,35,34,42,41,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,27)(14,20,21,34)(15,24,26,38)(17,28,30,40)(19,31,32,42)(22,33,29,41)(23,35,36,43)(25,37,39,44);(6,9)(10,12)(15,25)(17,30)(19,23)(22,29)(24,37)(26,39)(28,40)(31,35)(32,36)(33,41)(38,44)(42,43);(5,7,8,11)(6,10,9,12)(14,21)(16,27)(19,32)(22,29)(23,36)(24,38)(28,40)(37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056575);