#7312352

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1680 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)(17,18,22,28,23,27,19,24)(20,31,29,41,34,39,32,25)(21,26,35,42,37,44,30,38)(33,47,45,43,36,48,46,40);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,28)(20,32,34,29)(21,35,37,30)(25,39,41,31)(26,42,44,38)(33,45,36,46)(40,47,43,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,34)(18,25,27,41)(19,29,22,32)(21,36,37,33)(24,31,28,39)(26,43,44,40)(30,46,35,45)(38,48,42,47);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21)(18,26)(19,30)(20,33)(22,35)(23,37)(24,38)(25,40)(27,44)(28,42)(29,45)(31,47)(32,46)(34,36)(39,48)(41,43);(2,5)(6,8)(10,14)(13,15)(18,27)(21,37)(24,28)(25,41)(30,35)(31,39)(33,36)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7219548);