#6699697

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(19,25)(20,34)(21,27)(22,28)(23,29)(24,30)(26,37)(31,46)(32,41)(33,42)(35,48)(36,45)(38,44)(39,43)(40,47);(1,3)(2,5)(4,8)(6,7)(17,19,23,33)(18,25,29,42)(20,35,37,39)(21,32,24,40)(22,38,31,36)(26,43,34,48)(27,41,30,47)(28,44,46,45);(2,6)(5,7)(10,13)(14,16)(17,20,23,37)(18,26,29,34)(19,31,33,22)(21,38,24,36)(25,28,42,46)(27,45,30,44)(32,39,40,35)(41,48,47,43);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,30)(19,32,33,40)(20,36,37,38)(22,39,31,35)(25,41,42,47)(26,44,34,45)(28,43,46,48);(3,8)(5,7)(17,22,23,31)(18,28,29,46)(19,20,33,37)(21,39,24,35)(25,34,42,26)(27,43,30,48)(32,36,40,38)(41,45,47,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fg,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, 6606893);