Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9008 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #386 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1062 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);(3,4,9,12,8,5,7,11)(6,10,16,18,15,13,14,17)(19,20)(21,30,33,41)(22,34)(23,28)(24,38)(25,29)(26,32,42,37)(27,35)(31,46,49,40)(36,43)(39,44)(45,47,50,48);(3,5,8,4)(6,13,15,10)(7,11,9,12)(14,17,16,18)(19,21,24,32,25,33,23,37)(20,26,38,30,29,42,28,41)(22,31,39,47,36,49,35,48)(27,40,34,45,44,46,43,50);(3,6,8,15)(4,10,5,13)(7,16,9,14)(11,18,12,17)(19,22,25,36)(20,27,29,44)(21,31,33,49)(23,35,24,39)(26,40,42,46)(28,43,38,34)(30,45,41,50)(32,47,37,48)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101805);