#6452862

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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)(19,24)(20,25)(21,36)(22,26)(23,27)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,44)(35,43)(45,47)(46,48);(2,5)(6,8)(17,19,23,31)(18,24,27,40)(20,32,33,45)(21,30,22,29)(25,41,42,47)(26,38,36,39)(28,34,46,35)(37,44,48,43);(9,11,13,12)(10,14,15,16)(17,20,23,33)(18,25,27,42)(19,28,31,46)(21,35,22,34)(24,37,40,48)(26,44,36,43)(29,32,30,45)(38,41,39,47);(9,12,13,11)(10,14,15,16)(19,29)(20,33)(21,22)(24,38)(25,42)(26,36)(28,45)(30,31)(32,46)(37,47)(39,40)(41,48);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,27,36)(19,30,31,29)(20,34,33,35)(24,38,40,39)(25,43,42,44)(28,32,46,45)(37,47,48,41)

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,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, 6360058);