#6579523

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1287 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486719);