#6586125

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

1461 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)(17,18,22,26)(19,23,30,40)(20,24,33,43)(21,25,36,46)(27,37,28,38)(29,39,34,44)(31,41,47,48)(32,42,35,45);(1,3,5,4)(2,6,7,8)(17,19,22,30)(18,23,26,40)(20,31,33,47)(21,34,36,29)(24,41,43,48)(25,44,46,39)(27,35,28,32)(37,45,38,42);(9,11,12,16)(10,13,14,15)(17,20,22,33)(18,24,26,43)(19,27,30,28)(21,35,36,32)(23,37,40,38)(25,45,46,42)(29,47,34,31)(39,48,44,41);(1,4,5,3)(2,6,7,8)(20,32)(21,36)(24,42)(25,46)(27,31)(28,47)(29,34)(33,35)(37,41)(38,48)(39,44)(43,45);(17,20,22,33)(18,24,26,43)(19,28,30,27)(21,35,36,32)(23,38,40,37)(25,45,46,42)(29,31,34,47)(39,41,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,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, 6493321);