#6570513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 6477709);