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×#13500 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2076 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
678 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);(3,4,8,13,28,42,59,55,9,14,31,45,7,12,27,41)(5,15,18,47,17,36,52,43,20,35,29,63,19,10,32,34)(6,21,24,40,26,61,62,39,25,56,60,38,23,44,30,11)(16,48,50,33,49,54,64,65,22,57,58,37,51,66,53,46);(3,5)(4,10)(6,22)(7,19)(8,29)(9,20)(11,37)(12,35)(13,43)(14,36)(15,42)(16,25)(17,28)(18,31)(21,54)(23,49)(24,50)(26,51)(27,52)(30,53)(32,59)(33,39)(34,45)(38,65)(40,46)(41,47)(44,48)(55,63)(56,66)(57,61)(58,60)(62,64);(3,6,9,25)(4,11,14,39)(5,16)(7,26,28,23)(8,30,31,62)(10,33)(12,40,42,38)(13,44,45,61)(15,46)(17,51)(18,53)(19,49)(20,22)(21,55,56,41)(24,59,60,27)(29,64)(32,50)(34,48)(35,65)(36,37)(43,57)(47,66)(52,58)(54,63)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406595);