#6587429

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

1341 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)(17,18,22,27)(19,29,31,24)(20,33,36,25)(21,26,32,42)(23,28,38,41)(30,40,35,47)(34,43,48,45)(37,46,39,44);(1,3,5,4)(2,6,7,8)(17,19)(18,24)(20,34)(21,23)(22,31)(25,43)(26,41)(27,29)(28,42)(30,37)(32,38)(33,45)(35,39)(36,48)(40,44)(46,47);(10,13)(14,16)(17,20,22,36)(18,25,27,33)(19,30,31,35)(21,34,32,48)(23,39,38,37)(24,40,29,47)(26,45,42,43)(28,46,41,44);(9,11,12,15)(10,14,13,16)(17,21,22,32)(18,26,27,42)(19,23,31,38)(20,35,36,30)(24,41,29,28)(25,40,33,47)(34,39,48,37)(43,44,45,46);(1,4,5,3)(2,6,7,8)(19,21)(23,38)(24,42)(26,29)(28,41)(30,34)(31,32)(35,48)(37,39)(40,43)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,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, 6494625);