Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9335 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1070 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);(3,4,6,10,8,12,7,11)(5,13,14,9,15,17,16,18)(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)(4,9,12,18)(6,14)(7,16)(8,15)(10,17,11,13)(19,21,23,37,25,33,24,32)(20,26,28,41,29,42,38,30)(22,31,35,48,36,49,39,47)(27,40,43,50,44,46,34,45);(4,10)(6,7)(9,17)(11,12)(13,18)(14,16)(19,22)(20,27)(21,31)(23,35)(24,39)(25,36)(26,40)(28,43)(29,44)(30,45)(32,47)(33,49)(34,38)(37,48)(41,50)(42,46)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106761);