#6586140

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,27)(22,28)(23,26)(29,38)(30,40)(31,39)(32,41)(33,43)(34,45)(35,42)(36,46)(37,44)(47,48);(1,3,5,4)(2,6,7,8)(17,19,23,31)(18,24,26,39)(20,32,21,34)(22,36,37,30)(25,41,27,45)(28,46,44,40)(29,35,47,33)(38,42,48,43);(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,25,26,27)(19,29,31,47)(22,35,37,33)(24,38,39,48)(28,42,44,43)(30,34,36,32)(40,45,46,41);(1,4,5,3)(2,6,7,8)(18,26)(20,33)(21,35)(22,37)(24,39)(25,42)(27,43)(29,32)(30,36)(34,47)(38,45)(41,48);(17,21,23,20)(18,27,26,25)(19,29,31,47)(22,33,37,35)(24,38,39,48)(28,43,44,42)(30,34,36,32)(40,45,46,41)

Polycyclic presentation:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493336);