#6724924

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

1477 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)(11,14)(12,15)(13,16)(17,18,22,26)(19,23,30,39)(20,24,32,41)(21,25,35,44)(27,36,45,47)(28,37,29,38)(31,40,46,48)(33,42,34,43);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,16,15)(17,19,22,30)(18,23,26,39)(20,31,32,46)(21,28,35,29)(24,40,41,48)(25,37,44,38)(27,34,45,33)(36,43,47,42);(9,11,13,12)(10,15,16,14)(17,20)(18,24)(19,27)(21,34)(22,32)(23,36)(25,43)(26,41)(28,31)(29,46)(30,45)(33,35)(37,40)(38,48)(39,47)(42,44);(2,6)(5,7)(20,32)(24,41)(27,45)(31,46)(33,34)(36,47)(40,48)(42,43);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(19,28)(20,32)(21,35)(23,37)(24,41)(25,44)(27,46)(29,30)(31,45)(36,48)(38,39)(40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,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, 6632120);