#6448522

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,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, 6355718);