#7611259

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

1510 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 7518455);