Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #409371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15018 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15,19,9)(4,21)(5,27)(6,14)(7,13)(8,12)(10,24)(11,28)(16,42,22,51)(17,46,48,56)(18,33,20,34)(23,36)(25,37)(26,38)(29,40)(30,39)(31,44,35,45)(32,47,41,49)(43,58,52,57)(50,53)(54,59)(55,60)(61,63,64,62);(1,3,7,20,8,19,6,18)(2,9,13,33,12,15,14,34)(4,22,25,44,26,16,24,45)(5,17,28,47,29,48,30,49)(10,35,21,42,37,31,38,51)(11,32,40,46,39,41,27,56)(23,52,53,63,54,43,55,62)(36,58,50,61,59,57,60,64);(1,4,8,26)(2,10,12,37)(3,16,19,22)(5,23,29,54)(6,25,7,24)(9,31,15,35)(11,36,39,59)(13,38,14,21)(17,43,48,52)(18,45,20,44)(27,50,40,60)(28,55,30,53)(32,57,41,58)(33,42,34,51)(46,61,56,64)(47,63,49,62);(1,5)(2,11,12,39)(3,17)(4,23)(6,30)(7,28)(8,29)(9,32,15,41)(10,36,37,59)(13,40,14,27)(16,43)(18,49)(19,48)(20,47)(21,50,38,60)(22,52)(24,55)(25,53)(26,54)(31,57,35,58)(33,46,34,56)(42,61,51,64)(44,63)(45,62)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316568);