Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1258 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)(5,16)(6,22,25,40)(7,26)(8,14)(9,32)(10,15)(11,19)(12,24,39,31)(13,28)(17,42)(18,49,52,65)(20,44)(21,46)(23,38,27,41)(29,33)(30,45)(34,61,64,66)(35,55)(36,47)(37,56)(43,51,62,53)(48,58,59,60)(50,57,54,63);(3,5,17,47,9,20,7,19,10,21,28,55,8,30,29,56)(4,11,33,44,14,36,13,16,15,37,26,45,32,35,42,46)(6,18,48,61,24,53,27,54,25,52,59,66,31,51,23,50)(12,34,60,65,40,63,41,43,39,64,58,49,22,57,38,62);(3,6)(4,12)(5,18)(7,27)(8,31)(9,24)(10,25)(11,34)(13,41)(14,40)(15,39)(16,43)(17,48)(19,54)(20,53)(21,52)(22,32)(23,29)(26,58)(28,59)(30,51)(33,60)(35,57)(36,63)(37,64)(38,42)(44,65)(45,49)(46,62)(47,61)(50,56)(55,66)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401846);