#6693541

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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6600737);