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×#8476 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2190 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);(3,4,6,10,7,8,14,16)(5,11,12,18,13,9,15,17)(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);(4,8)(6,14)(9,11)(12,15)(19,21)(20,25)(22,30)(23,37)(24,32)(26,40)(27,43)(28,41)(29,36)(31,39)(33,49)(34,48)(35,47)(38,46)(42,45)(44,50);(3,5,7,13)(4,9,8,11)(6,15,14,12)(10,18,16,17)(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, 94300);