#6579567

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

1400 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,42)(28,41)(29,37)(30,43)(31,44)(32,38)(33,45)(34,39)(35,40)(36,46)(47,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(22,34)(24,40)(25,38)(28,31)(29,35)(30,47)(33,36)(37,42)(41,43)(44,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19,21,29)(18,24,23,37)(20,30,32,31)(22,35,34,27)(25,43,38,44)(26,40,39,42)(28,36,47,33)(41,46,48,45);(17,20,21,32)(18,25,23,38)(19,28,29,47)(22,36,34,33)(24,41,37,48)(26,46,39,45)(27,31,35,30)(40,43,42,44);(17,19,21,29)(18,24,23,37)(20,31,32,30)(22,35,34,27)(25,44,38,43)(26,40,39,42)(28,33,47,36)(41,45,48,46)

Polycyclic presentation:

2^9|b^2=fgi,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,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, 6486763);