#7312495

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

1685 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)(17,18,23,27,22,28,21,29)(19,30,33,42,34,24,35,45)(20,36,37,25,38,26,39,43)(31,44,48,46,32,41,47,40);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,34)(18,24,28,30)(20,31,38,32)(21,35,23,33)(25,40,43,46)(26,44,36,41)(27,45,29,42)(37,48,39,47);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,31)(21,37)(22,38)(23,39)(24,40)(26,29)(27,36)(28,43)(30,46)(32,34)(33,47)(35,48)(41,45)(42,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,26)(19,32)(21,39)(22,38)(23,37)(24,41)(25,29)(27,43)(28,36)(30,44)(31,34)(33,47)(35,48)(40,45)(42,46);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,33,34,35)(20,37,38,39)(24,42,30,45)(25,36,43,26)(31,47,32,48)(40,44,46,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,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, 7219691);