#6713560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1566 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)(9,10,12,13)(11,14,15,16)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(45,48)(46,47);(1,3)(2,5)(4,8)(6,7)(17,19,23,32)(18,24,27,40)(20,33,35,46)(21,31,22,30)(25,41,43,47)(26,39,28,38)(29,36,45,34)(37,44,48,42);(10,13)(14,16)(17,20,23,35)(18,25,27,43)(19,29,32,45)(21,34,22,36)(24,37,40,48)(26,42,28,44)(30,33,31,46)(38,41,39,47);(2,6)(5,7)(17,21,23,22)(18,26,27,28)(19,30,32,31)(20,34,35,36)(24,38,40,39)(25,42,43,44)(29,33,45,46)(37,41,48,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,27)(19,30)(21,22)(24,39)(25,43)(29,33)(31,32)(34,36)(37,47)(38,40)(41,48)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6620756);