#6580841

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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)(20,24,34,42)(22,26,35,43)(28,38,29,37)(30,41,33,39)(32,40,36,44)(45,48,46,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(19,28)(20,32)(22,35)(23,37)(24,40)(26,43)(27,38)(29,31)(34,36)(42,44)(45,46)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19)(18,23)(20,33)(21,31)(22,29)(24,41)(25,27)(26,38)(28,35)(30,34)(32,46)(36,45)(37,43)(39,42)(40,48)(44,47);(1,4,5,3)(2,8,6,7)(19,29)(20,32)(22,35)(23,38)(24,40)(26,43)(27,37)(28,31)(30,33)(34,36)(39,41)(42,44);(17,20,21,34)(18,24,25,42)(19,30,31,33)(22,36,35,32)(23,39,27,41)(26,44,43,40)(28,45,29,46)(37,47,38,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,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, 6488037);