Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8478 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,7,12,8,11,15,18)(5,10,13,17,6,9,14,16)(19,20)(21,29)(22,33,35,26)(23,36)(24,28)(25,31)(27,39)(30,45,47,50)(32,43)(34,42,38,44)(37,41)(40,46,49,48);(3,5,8,6)(4,9,11,10)(7,14,15,13)(12,17,18,16)(19,21,24,32)(20,25,28,41)(22,30,35,47)(23,37,39,31)(26,40,33,49)(27,43,36,29)(34,48,38,46)(42,45,44,50);(3,6,8,5)(4,10,11,9)(7,13,15,14)(12,16,18,17)(19,22,24,35)(20,26,28,33)(21,30,32,47)(23,38,39,34)(25,40,41,49)(27,44,36,42)(29,45,43,50)(31,48,37,46)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94301);