#6580237

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

1498 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=i,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, 6487433);