#6570442

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,31,35)(27,36,38,49)(28,40,44,55)(29,34,30,46)(32,45,48,39)(33,42,51,43)(37,50,41,47)(52,56,54,53);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,45,31,38,30,39)(26,32,46,49,35,48,34,36)(28,41,42,53,44,37,43,56)(33,50,55,52,51,47,40,54);(11,17)(14,20)(18,23)(21,24)(25,28,30,43,31,44,29,42)(26,33,34,40,35,51,46,55)(27,37,39,53,38,41,45,56)(32,47,36,52,48,50,49,54);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(32,48)(36,49)(37,41)(39,45)(47,50)(52,54)(53,56);(1,4,5,3)(2,6,8,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,46)(27,39,38,45)(28,42,44,43)(32,49,48,36)(33,40,51,55)(37,53,41,56)(47,54,50,52)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=fgi,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,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477638);