#7614733

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,25)(21,36)(22,27)(23,28)(24,31)(26,38)(30,46)(32,42)(33,41)(34,43)(35,44)(37,45)(39,40)(47,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,29)(20,30,34,39)(21,35,38,47)(23,33,37,32)(25,40,43,46)(26,44,36,48)(28,42,45,41);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,34)(18,25,27,43)(19,30,31,39)(21,33,38,32)(23,35,37,47)(24,40,29,46)(26,42,36,41)(28,44,45,48);(10,14)(13,15)(17,21,37,30,22,38,23,39)(18,26,45,40,27,36,28,46)(19,20,33,47,31,34,32,35)(24,25,42,48,29,43,41,44);(2,5)(6,8)(10,14)(13,15)(18,27)(20,34)(21,38)(24,29)(28,45)(30,39)(35,47)(41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=hi,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, 7521929);