#6450268

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(9,10,12,13)(11,14,15,16)(17,18)(19,24)(20,26)(21,35)(22,25)(23,27)(28,44)(29,38)(30,39)(31,37)(32,41)(33,43)(34,40)(36,42)(45,48)(46,47);(2,5)(6,8)(17,19,22,30)(18,24,25,39)(20,32,34,46)(21,31,23,28)(26,41,40,47)(27,44,35,37)(29,36,45,33)(38,42,48,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,25)(19,28)(21,23)(24,37)(26,40)(29,32)(30,31)(33,36)(38,47)(39,44)(41,48)(45,46);(9,11,12,15)(10,14,13,16)(17,20,22,34)(18,26,25,40)(19,29,30,45)(21,33,23,36)(24,38,39,48)(27,42,35,43)(28,32,31,46)(37,47,44,41);(17,21,22,23)(18,27,25,35)(19,28,30,31)(20,33,34,36)(24,37,39,44)(26,42,40,43)(29,32,45,46)(38,47,48,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6357464);