#6699715

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,33,42)(20,26,37,46)(21,27,23,29)(22,28,31,40)(32,41,39,48)(34,43,38,47)(35,44,36,45);(1,3)(2,5)(4,8)(6,7)(17,19,24,33)(18,25,30,42)(20,34,37,38)(21,32,23,39)(22,36,31,35)(26,43,46,47)(27,41,29,48)(28,45,40,44);(2,6)(5,7)(10,13)(14,16)(17,20,24,37)(18,26,30,46)(19,31,33,22)(21,36,23,35)(25,40,42,28)(27,45,29,44)(32,38,39,34)(41,47,48,43);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,24,23)(18,27,30,29)(19,32,33,39)(20,35,37,36)(22,38,31,34)(25,41,42,48)(26,44,46,45)(28,47,40,43);(3,8)(5,7)(17,22,24,31)(18,28,30,40)(19,20,33,37)(21,38,23,34)(25,26,42,46)(27,47,29,43)(32,35,39,36)(41,44,48,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fgi,e^2=i,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, 6606911);