Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1942 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);(3,4,9,15,10,16,8,14)(5,17,22,47,23,48,21,46)(6,24,27,12,28,40,26,41)(7,29,18,44,32,13,33,35)(11,34,37,30,38,56,43,20)(19,50,53,45,54,65,52,62)(25,59,49,39,31,58,60,42)(36,55,63,66,64,51,57,61);(3,5,18,34)(4,11,35,47)(6,19,49,61)(7,30,9,21)(8,22,32,20)(10,23,33,56)(12,36,58,50)(13,17,15,37)(14,43,29,48)(16,38,44,46)(24,57,42,45)(25,55,27,52)(26,53,31,51)(28,54,60,66)(39,62,40,63)(41,64,59,65);(3,6,10,28)(4,12,16,41)(5,19,23,54)(7,31,32,25)(8,27,9,26)(11,36,38,64)(13,42,29,39)(14,40,15,24)(17,45,48,62)(18,49,33,60)(20,55,30,51)(21,53,22,52)(34,61,56,66)(35,58,44,59)(37,57,43,63)(46,65,47,50)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401951);