Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10779 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #379 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8)(9,18)(10,22,24,31)(11,25)(12,16)(13,17)(14,20)(15,23,32,27)(19,34,37,28)(21,29)(26,30)(33,36,38,35);(3,4,5,6)(7,9)(8,14)(10,19)(11,26)(12,20)(13,21)(15,28)(16,29)(17,30)(18,25)(22,33)(23,36)(24,37)(27,35)(31,38)(32,34);(7,10)(8,15)(9,19)(11,23)(12,27)(13,24)(14,28)(16,31)(17,32)(18,33)(20,35)(21,37)(22,25)(26,36)(29,38)(30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260649);