#6551967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459163);