#6729907

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=gi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6637103);