#6729282

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	#42906	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,21)(19,24,31,28)(20,25,33,35)(22,26,38,36)(27,39,46,47)(29,40,30,41)(32,42,45,48)(34,43,37,44);(1,3)(2,5)(4,8)(6,7)(17,19,23,31)(18,24,21,28)(20,32,33,45)(22,29,38,30)(25,42,35,48)(26,40,36,41)(27,37,46,34)(39,44,47,43);(10,14)(15,16)(17,20,23,33)(18,25,21,35)(19,27,31,46)(22,37,38,34)(24,39,28,47)(26,44,36,43)(29,32,30,45)(40,42,41,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,18)(19,28,31,24)(20,25,33,35)(22,36,38,26)(27,39,46,47)(29,41,30,40)(32,42,45,48)(34,43,37,44);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,29)(20,33)(22,38)(24,40)(25,35)(26,36)(27,45)(28,41)(30,31)(32,46)(39,48)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6636478);