#7202109

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,28)(29,38)(31,44)(32,40)(33,37)(34,41)(35,42)(39,43)(45,47)(46,48);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,29,35,34)(25,37,40,30)(26,38,42,41)(31,43,45,46)(39,47,48,44);(11,15)(13,16)(17,20,23,33)(18,25,27,40)(19,28,22,32)(21,31,35,45)(24,37,36,30)(26,39,42,48)(29,43,34,46)(38,47,41,44);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,21,23,35)(18,26,27,42)(19,29,22,34)(20,31,33,45)(24,38,36,41)(25,39,40,48)(28,43,32,46)(30,44,37,47);(2,5)(6,8)(11,15)(13,16)(17,19,23,22)(18,24,27,36)(20,32,33,28)(21,34,35,29)(25,30,40,37)(26,41,42,38)(31,43,45,46)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7109305);