Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1274 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);(3,4,8,14,33,36,31,13,10,16,30,47,9,15,7,29)(5,17,20,11,34,38,19,48,22,51,58,40,21,45,32,39)(6,23,26,44,35,46,24,43,28,41,61,12,27,42,25,49)(18,37,54,50,59,66,53,62,56,65,57,60,55,64,52,63);(3,5,9,34,10,22,33,21)(4,11,36,39,16,40,15,48)(6,18,35,55,28,56,27,59)(7,20,30,19,31,58,8,32)(12,37,43,66,44,65,49,64)(13,45,47,51,29,38,14,17)(23,60,42,63,41,50,46,62)(24,57,61,53,25,54,26,52);(3,6)(4,12)(5,18)(7,26)(8,25)(9,35)(10,28)(11,37)(13,46)(14,41)(15,49)(16,44)(17,50)(19,57)(20,52)(21,59)(22,56)(23,47)(24,30)(27,33)(29,42)(31,61)(32,54)(34,55)(36,43)(38,63)(39,66)(40,65)(45,62)(48,64)(51,60)(53,58)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401744);