Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11320 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1502 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);(4,5)(7,8)(9,17,21,31)(10,22)(11,15)(12,16)(13,20,29,26)(14,25)(18,27)(19,32,23,37)(24,36)(28,34,35,38)(30,33);(3,4)(5,6)(7,9,18,20,12,21,11,26)(8,13,27,17,16,29,15,31)(10,23,33,38,25,19,24,34)(14,28,36,32,22,35,30,37);(3,5,6,4)(7,10,11,24,12,25,18,33)(8,14,15,30,16,22,27,36)(9,19,20,34,21,23,26,38)(13,28,17,32,29,35,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265564);