#6551951

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

Polycyclic presentation:

2^9|a^2=g,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, 6459147);