Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#12897 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,16,22,15,21,31,38,17,23,33,19,32,20,14,29)(13,24,27,39,26,18,25,34,28,35,40,42,41,36,30,37);(3,5,6,9)(4,7,8,10)(11,13,17,28)(12,18)(14,30,31,25)(15,26,32,41)(16,27,33,40)(19,37)(20,24)(21,35)(22,34)(23,36)(29,39)(38,42);(4,8)(7,10)(11,14,15,16,17,31,32,33)(12,19,20,38,23,22,21,29)(13,25,26,40,28,30,41,27)(18,34,24,39,36,37,35,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401297);