#6452830

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

1635 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)(19,27)(20,24)(21,25)(22,26)(23,30)(28,45)(29,39)(31,38)(32,48)(33,41)(34,42)(35,43)(36,44)(37,47)(40,46);(2,5)(6,8)(17,19,21,30)(18,23,25,27)(20,32,33,46)(22,29,35,31)(24,40,41,48)(26,38,43,39)(28,36,47,34)(37,44,45,42);(9,11,13,12)(10,14,15,16)(17,20,21,33)(18,24,25,41)(19,28,30,47)(22,36,35,34)(23,37,27,45)(26,44,43,42)(29,32,31,46)(38,40,39,48);(9,12,13,11)(10,14,15,16)(19,29)(20,33)(22,35)(23,38)(24,41)(26,43)(27,39)(28,46)(30,31)(32,47)(37,48)(40,45);(1,3,4,7)(2,6,5,8)(18,25)(20,33)(23,27)(26,43)(28,47)(32,46)(34,36)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6360026);