#6580909

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,31,23)(20,32,34,24)(22,26,35,41)(28,38,29,37)(30,39,46,43)(33,42,36,40)(44,47,45,48);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(19,28)(20,33)(22,35)(23,37)(24,40)(26,41)(27,38)(29,31)(32,42)(34,36)(44,45)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,21,31)(18,23,25,27)(20,30,34,46)(22,29,35,28)(24,39,32,43)(26,38,41,37)(33,44,36,45)(40,47,42,48);(1,4,5,3)(2,8,6,7)(19,29)(20,33)(22,35)(23,38)(24,40)(26,41)(27,37)(28,31)(30,46)(32,42)(34,36)(39,43);(17,20,21,34)(18,24,25,32)(19,30,31,46)(22,36,35,33)(23,39,27,43)(26,42,41,40)(28,44,29,45)(37,47,38,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6488105);