#6580228

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(17,18,22,27)(19,24,30,40)(20,31,33,25)(21,26,36,44)(23,28,37,45)(29,47,48,39)(32,43,35,41)(34,46,38,42);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,30)(18,24,27,40)(20,32,33,35)(21,23,36,37)(25,41,31,43)(26,28,44,45)(29,38,48,34)(39,46,47,42);(10,14)(13,16)(17,20,22,33)(18,25,27,31)(19,29,30,48)(21,35,36,32)(23,38,37,34)(24,39,40,47)(26,43,44,41)(28,46,45,42);(1,3,4,7)(2,6,5,8)(17,21)(18,26)(19,23)(20,29)(22,36)(24,28)(25,39)(27,44)(30,37)(31,47)(32,38)(33,48)(34,35)(40,45)(41,46)(42,43);(19,21)(23,37)(24,26)(28,45)(29,35)(30,36)(32,48)(34,38)(39,43)(40,44)(41,47)(42,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,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, 6487424);