Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10760 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4058 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);(3,4)(5,6)(7,8,11,18,13,17,12,16)(9,19,21,14,23,30,22,29)(10,24,26,32,27,15,25,31)(20,34,36,28,37,33,35,38);(4,6)(7,9,12,22,13,23,11,21)(8,14,18,30,17,29,16,19)(10,20,26,36,27,37,25,35)(15,28,32,34,24,38,31,33);(7,10)(8,15)(9,20)(11,25)(12,26)(13,27)(14,28)(16,31)(17,24)(18,32)(19,33)(21,35)(22,36)(23,37)(29,38)(30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260523);