#6590747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

1471 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497943);