Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11288 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #385 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1678 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);(7,8,11,16,12,15,24,30)(9,13,19,27,18,26,22,31)(10,14,21,29,20,28,23,32)(17,25,34,36,33,37,35,38);(3,4)(5,6)(7,9)(8,13,15,26)(10,17,20,33)(11,22)(12,18)(14,25)(16,31,30,27)(19,24)(21,35,23,34)(28,37)(29,38)(32,36);(4,6)(7,10)(8,14,15,28)(9,17,18,33)(11,23)(12,20)(13,25)(16,32,30,29)(19,35,22,34)(21,24)(26,37)(27,38)(31,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265451);