Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23274 |
| prime radical | #575 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,9,12,35,28,7,13,34,18,6,32,16,45)(3,15,21,44,22,30,42,53,20,33,41,47,19,5,11,31)(4,10,25,38,26,37,58,48,27,39,59,54,24,56,46,63)(17,43,50,62,51,57,61,64,52,23,36,55,49,29,40,60);(1,3,16,11,6,19,34,41,7,20,35,42,9,22,8,21)(2,5,28,44,32,33,14,31,13,30,45,47,12,15,18,53)(4,17,46,40,24,49,59,36,27,52,58,61,26,51,25,50)(10,29,48,62,56,23,38,60,39,57,63,55,37,43,54,64);(1,4,6,24,7,27,9,26)(2,10,12,37,13,39,32,56)(3,17,19,49,20,52,22,51)(5,29,15,43,30,57,33,23)(8,25,16,46,34,59,35,58)(11,40,41,36,42,61,21,50)(14,38,28,48,18,54,45,63)(31,60,44,62,53,64,47,55);(1,5,7,30)(2,11,13,42)(3,18,20,14)(4,23)(6,33,9,15)(8,31,34,53)(10,36)(12,21,32,41)(16,47,35,44)(17,48)(19,28,22,45)(24,57)(25,55)(26,29)(27,43)(37,40)(38,49)(39,50)(46,64)(51,54)(52,63)(56,61)(58,60)(59,62)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=dei,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405868);