Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13489 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2073 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4)(5,17,21,11)(6,22,26,46)(7,14)(8,15)(9,16)(10,13)(12,24,38,33)(18,49)(19,45,32,35)(20,48,54,36)(23,56)(25,43,59,66)(27,41)(28,60,61,42)(29,62,63,39)(30,44)(31,40)(34,50)(37,57)(47,51)(52,64)(53,65)(55,58);(3,5)(4,11)(6,23)(7,19)(8,28)(9,32)(10,21)(12,37)(13,17)(14,35)(15,42)(16,45)(18,25)(20,30)(22,55)(24,57)(26,58)(27,54)(29,50)(31,61)(33,53)(34,39)(36,44)(38,65)(40,60)(41,48)(43,64)(46,56)(47,62)(49,66)(51,63)(52,59);(3,6)(4,12,13,38)(5,18)(7,24)(8,29)(9,33)(10,26)(11,34,17,47)(14,22,16,46)(15,43,40,66)(19,50)(20,53)(21,52)(23,28)(25,30)(27,59)(31,63)(32,51)(35,64,45,49)(36,56,48,55)(37,60,65,42)(39,41,62,44)(54,57)(58,61)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406561);