#6452715

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 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)(11,16)(12,14)(13,15)(17,18,21,25)(19,23,29,39)(20,31,33,24)(22,26,35,43)(27,45,47,37)(28,38,30,40)(32,48,46,41)(34,44,36,42);(2,5)(6,8)(17,19,21,29)(18,23,25,39)(20,32,33,46)(22,28,35,30)(24,41,31,48)(26,38,43,40)(27,36,47,34)(37,44,45,42);(9,11,13,12)(10,14,15,16)(17,20,21,33)(18,24,25,31)(19,27,29,47)(22,36,35,34)(23,37,39,45)(26,44,43,42)(28,32,30,46)(38,41,40,48);(9,12,13,11)(10,14,15,16)(19,28)(20,33)(22,35)(23,38)(24,31)(26,43)(27,46)(29,30)(32,47)(37,48)(39,40)(41,45);(1,3,4,7)(2,6,5,8)(20,33)(24,31)(27,47)(32,46)(34,36)(37,45)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^c=hi,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, 6359911);