#6448558

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,29,32,24)(20,25,36,43)(21,26,23,28)(30,40,33,38)(31,45,47,39)(34,48,46,41)(35,42,37,44);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,32)(18,24,27,29)(20,34,36,46)(21,33,23,30)(25,41,43,48)(26,40,28,38)(31,35,47,37)(39,42,45,44);(9,12,13,11)(10,14,15,16)(19,30)(21,23)(24,38)(26,28)(29,40)(31,46)(32,33)(34,47)(35,37)(39,48)(41,45)(42,44);(17,20,22,36)(18,25,27,43)(19,31,32,47)(21,35,23,37)(24,39,29,45)(26,42,28,44)(30,46,33,34)(38,48,40,41);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,28)(19,30,32,33)(20,35,36,37)(24,38,29,40)(25,42,43,44)(31,46,47,34)(39,48,45,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,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, 6355754);