#6582783

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,28,21,24)(20,32,34,25)(23,27,37,45)(29,39,36,47)(30,41,31,40)(33,42,48,43)(35,46,38,44);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,22,21)(18,24,26,28)(20,33,34,48)(23,30,37,31)(25,42,32,43)(27,40,45,41)(29,38,36,35)(39,46,47,44);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20,22,34)(18,25,26,32)(19,29,21,36)(23,38,37,35)(24,39,28,47)(27,46,45,44)(30,33,31,48)(40,42,41,43);(1,3,5,4)(2,7,8,6)(17,21)(18,24)(19,22)(20,33)(23,31)(25,43)(26,28)(27,40)(29,35)(30,37)(32,42)(34,48)(36,38)(39,46)(41,45)(44,47);(1,4,5,3)(2,6,8,7)(19,30)(21,31)(23,37)(24,40)(27,45)(28,41)(29,33)(35,38)(36,48)(39,42)(43,47)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=fi,e^2=fi,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, 6489979);