#7233763

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

Polycyclic presentation:

2^9|c^2=h,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7140959);