#6570988

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,35,31,34,30,36)(27,37,39,32,41,48,40,47)(28,42,45,33,44,50,43,49)(38,46,54,51,53,55,52,56);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,31,41)(26,32,34,47)(28,38,44,53)(29,40,30,39)(33,46,49,55)(35,37,36,48)(42,51,50,56)(43,54,45,52);(1,3,5,4)(2,7,8,6)(10,15)(11,18)(16,22)(19,24)(25,28,31,44)(26,33,34,49)(27,38,41,53)(29,43,30,45)(32,46,47,55)(35,42,36,50)(37,51,48,56)(39,52,40,54);(1,4,5,3)(2,6,8,7)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(26,34)(32,47)(33,49)(35,36)(37,48)(42,50)(46,55)(51,56);(12,20)(16,22)(19,24)(21,23)(25,29,31,30)(26,35,34,36)(27,39,41,40)(28,43,44,45)(32,48,47,37)(33,42,49,50)(38,52,53,54)(46,56,55,51)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 6478184);