#511707

order	512
nilpotency class	4
isoclinism	isoclinic class of #511571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

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, 418903);