#6579619

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 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)(17,18,21,25)(19,27,31,23)(20,24,34,42)(22,26,36,44)(28,39,30,37)(29,40,32,38)(33,41,35,43)(45,48,46,47);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,32)(21,31)(22,28)(24,40)(25,27)(26,37)(29,34)(30,36)(33,46)(35,45)(38,42)(39,44)(41,48)(43,47);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,28)(20,33)(22,36)(23,37)(24,41)(26,44)(27,39)(30,31)(34,35)(42,43)(45,46)(47,48);(17,20)(18,24)(19,29)(21,34)(22,33)(23,38)(25,42)(26,41)(27,40)(28,45)(30,46)(31,32)(35,36)(37,47)(39,48)(43,44);(19,30)(20,33)(22,36)(23,39)(24,41)(26,44)(27,37)(28,31)(29,32)(34,35)(38,40)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fi,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, 6486815);