#7611283

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1508 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)(17,18,22,27)(19,28,31,24)(20,25,35,45)(21,36,23,26)(29,43,33,39)(30,40,37,41)(32,48,47,42)(34,46,38,44);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,27,28)(20,32,35,47)(21,30,23,37)(25,42,45,48)(26,40,36,41)(29,38,33,34)(39,44,43,46);(10,14)(11,15)(17,19)(18,24)(20,33)(21,37)(22,31)(23,30)(25,43)(26,41)(27,28)(29,35)(32,38)(34,47)(36,40)(39,45)(42,44)(46,48);(10,14)(13,16)(17,20,21,38,22,35,23,34)(18,25,36,44,27,45,26,46)(19,29,37,47,31,33,30,32)(24,39,40,48,28,43,41,42);(1,3,4,8)(2,6,5,7)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,34,35,38)(24,40,28,41)(25,44,45,46)(29,47,33,32)(39,42,43,48)

Polycyclic presentation:

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

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, 7518479);