Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11240 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1818 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);(7,8,12,16,13,17,11,27)(9,18,21,29,22,32,20,14)(10,23,26,38,25,15,24,30)(19,33,36,28,35,31,34,37);(3,4)(5,6)(7,9,11,20,13,22,12,21)(8,14,16,18,17,29,27,32)(10,19,24,34,25,35,26,36)(15,28,30,31,23,37,38,33);(4,6)(7,10,11,24,13,25,12,26)(8,15,16,30,17,23,27,38)(9,19,20,34,22,35,21,36)(14,28,18,31,29,37,32,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265252);