#6590145

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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)(11,14)(12,16)(13,15)(17,18,23,28)(19,24,21,26)(20,25,34,43)(22,27,37,46)(29,38,35,44)(30,39,31,40)(32,41,47,48)(33,42,36,45);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,21)(18,24,28,26)(20,32,34,47)(22,30,37,31)(25,41,43,48)(27,39,46,40)(29,36,35,33)(38,45,44,42);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(17,20,23,34)(18,25,28,43)(19,29,21,35)(22,36,37,33)(24,38,26,44)(27,45,46,42)(30,32,31,47)(39,41,40,48);(9,12,13,11)(10,16,15,14)(17,21)(18,26)(19,23)(20,32)(22,31)(24,28)(25,41)(27,40)(29,33)(30,37)(34,47)(35,36)(38,42)(39,46)(43,48)(44,45);(1,4,5,3)(2,8,6,7)(19,30)(21,31)(22,37)(24,39)(26,40)(27,46)(29,32)(33,36)(35,47)(38,41)(42,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6497341);