#6442159

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,41)(29,43)(31,38)(32,39)(33,40)(34,36)(35,42)(37,45)(44,48)(46,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(19,28)(20,31)(22,33)(23,36)(24,38)(26,40)(27,41)(30,34)(32,35)(39,42)(44,46)(47,48);(10,14)(13,16)(17,19,21,30)(18,23,25,27)(20,29,32,45)(22,34,33,28)(24,37,39,43)(26,41,40,36)(31,44,35,46)(38,47,42,48);(17,20,21,32)(18,24,25,39)(19,29,30,45)(22,35,33,31)(23,37,27,43)(26,42,40,38)(28,44,34,46)(36,47,41,48);(1,3,4,7)(2,6,5,8)(18,25)(20,32)(23,27)(26,40)(29,45)(31,35)(36,41)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349355);