Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7729 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,11,13,8)(6,14)(7,10)(9,15)(12,16,18,17)(19,20)(21,27,29,34)(22,26)(23,25)(24,28,32,30)(31,33)(35,36,38,42,39,41,47,49)(37,43,45,50,44,40,46,48);(3,5)(4,8)(6,12)(7,13)(9,16)(10,11)(14,17)(15,18)(19,21,23,29)(20,24,25,32)(22,30,31,28)(26,34,33,27)(35,37)(36,40)(38,46)(39,44)(41,43)(42,50)(45,47)(48,49);(3,6,7,15)(4,9,10,14)(5,12,13,18)(8,16,11,17)(20,25)(24,32)(26,33)(27,34)(36,41)(37,44)(42,49)(45,46)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68897);