#6590940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498136);