#7312234

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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

1705 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)(17,18,23,36,22,27,21,26)(19,28,35,24,29,43,31,38)(20,25,37,48,32,40,34,42)(30,44,47,39,33,46,45,41);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(19,29)(20,32)(24,38)(25,40)(28,43)(31,35)(34,37)(42,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,24,27,38)(20,33,32,30)(21,35,23,31)(25,41,40,39)(26,43,36,28)(34,45,37,47)(42,44,48,46);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,32)(18,25,27,40)(19,30,29,33)(21,34,23,37)(24,39,38,41)(26,42,36,48)(28,44,43,46)(31,45,35,47);(2,5)(6,8)(10,14)(13,15)(20,32)(25,40)(30,33)(34,37)(39,41)(42,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7219430);