#7615213

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7522409);