#6678165

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2252 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,37)(20,24,32,41)(21,25,33,42)(27,36,34,43)(29,38,31,40)(30,39,35,44)(45,47,46,48);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(19,27)(20,30)(21,33)(23,36)(24,39)(25,42)(28,34)(32,35)(37,43)(41,44)(45,46)(47,48);(10,14)(13,16)(17,19)(18,23)(20,31)(21,34)(22,28)(24,40)(25,43)(26,37)(27,33)(29,32)(30,46)(35,45)(36,42)(38,41)(39,48)(44,47);(2,6)(5,7)(19,28)(23,37)(27,34)(29,31)(36,43)(38,40)(45,46)(47,48);(3,8)(5,7)(17,20,22,32)(18,24,26,41)(19,29,28,31)(21,35,33,30)(23,38,37,40)(25,44,42,39)(27,45,34,46)(36,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 6585361);