#7312624

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

1676 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)(17,18,23,38,22,27,19,24)(20,30,28,25,33,39,31,42)(21,34,35,40,37,26,29,43)(32,48,45,44,36,47,46,41);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,38)(20,31,33,28)(21,35,37,29)(25,30,42,39)(26,43,34,40)(32,45,36,46)(41,48,44,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,33)(18,25,27,42)(19,28,23,31)(21,36,37,32)(24,39,38,30)(26,44,34,41)(29,46,35,45)(40,47,43,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21)(18,26)(19,29)(20,32)(22,37)(23,35)(24,40)(25,41)(27,34)(28,45)(30,47)(31,46)(33,36)(38,43)(39,48)(42,44);(2,5)(6,8)(10,14)(13,15)(18,27)(21,37)(24,38)(25,42)(29,35)(30,39)(32,36)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 7219820);