#7233779

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,26)(21,28)(22,27)(23,29)(25,34)(30,38)(31,40)(32,39)(33,41)(35,43)(36,42)(37,44)(45,48)(46,47);(2,5)(3,7)(10,14)(11,12)(17,19,23,33)(18,24,29,41)(20,30,35,45)(21,31,22,32)(25,37,42,47)(26,38,43,48)(27,39,28,40)(34,44,36,46);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,37)(26,27)(28,43)(29,42)(31,44)(32,46)(33,45)(38,39)(40,48)(41,47);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,38)(25,27)(28,42)(29,43)(31,46)(32,44)(33,45)(37,39)(40,47)(41,48);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,29,28)(19,31,33,32)(20,34,35,36)(24,39,41,40)(25,26,42,43)(30,44,45,46)(37,38,47,48)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7140975);