#6587601

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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 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)(11,14)(12,13)(15,16)(17,18)(19,28)(20,33)(21,26)(22,27)(23,25)(24,31)(29,43)(30,38)(32,39)(34,42)(35,41)(36,40)(37,45)(44,47)(46,48);(1,3,5,4)(2,6,7,8)(17,19)(18,24)(20,32)(21,29)(22,31)(23,30)(25,39)(26,37)(27,28)(33,38)(34,44)(35,45)(36,46)(40,47)(41,43)(42,48);(10,13)(14,16)(17,20)(18,25)(22,23)(24,28)(26,41)(27,33)(29,44)(30,32)(34,36)(37,47)(43,48)(45,46);(9,11,12,15)(10,14,13,16)(17,21,22,35)(18,26,27,41)(19,29,31,45)(20,34,23,36)(24,37,28,43)(25,40,33,42)(30,46,32,44)(38,48,39,47);(1,4,5,3)(2,6,7,8)(18,27)(19,30)(20,23)(21,34)(24,38)(26,40)(28,39)(29,45)(31,32)(35,36)(41,42)(47,48)

Polycyclic presentation:

2^9|b^2=fi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=ehi,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, 6494797);