#6580141

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 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)(17,18,21,25)(19,27,31,23)(20,24,33,40)(22,26,34,41)(28,37,29,36)(30,43,46,38)(32,39,35,42)(44,48,45,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(19,28)(20,32)(22,34)(23,36)(24,39)(26,41)(27,37)(29,31)(33,35)(40,42)(44,45)(47,48);(10,14)(13,16)(17,19,21,31)(18,23,25,27)(20,30,33,46)(22,29,34,28)(24,38,40,43)(26,37,41,36)(32,44,35,45)(39,47,42,48);(1,3,4,7)(2,6,5,8)(19,29)(20,32)(22,34)(23,37)(24,39)(26,41)(27,36)(28,31)(30,46)(33,35)(38,43)(40,42);(17,20,21,33)(18,24,25,40)(19,30,31,46)(22,35,34,32)(23,38,27,43)(26,42,41,39)(28,44,29,45)(36,47,37,48)

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=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, 6487337);