#7508560

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1453 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,34)(22,26)(23,27)(28,37)(29,40)(30,39)(31,41)(32,44)(33,43)(35,38)(36,42)(45,48)(46,47);(1,3)(2,6)(4,7)(5,8)(17,19)(18,24)(20,31)(21,29)(22,35)(23,30)(25,41)(26,38)(27,39)(28,32)(33,45)(34,40)(36,46)(37,44)(42,47)(43,48);(9,11,12,16)(10,13,14,15)(17,20,23,33)(18,25,27,43)(19,28,30,46)(21,32,22,36)(24,37,39,47)(26,42,34,44)(29,45,35,31)(38,41,40,48);(2,7)(3,8)(10,14)(13,15)(17,21,23,22)(18,26,27,34)(19,29,30,35)(20,32,33,36)(24,38,39,40)(25,42,43,44)(28,45,46,31)(37,41,47,48);(1,4,5,6)(2,3,7,8)(18,27)(24,39)(25,43)(26,34)(37,47)(38,40)(41,48)(42,44)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7415756);