Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11600 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,22,25,38)(14,18)(15,29)(16,21)(17,24,33,41)(19,27)(20,28)(23,36,26,39)(30,32,40,34)(31,35)(37,42);(11,13,16,25)(12,17,21,33)(14,26,28,23)(15,30,31,40)(18,34,20,32)(19,36,37,39)(22,29,38,35)(24,27,41,42);(11,14,16,28)(12,18,21,20)(13,23,25,26)(15,27,31,42)(17,32,33,34)(19,35,37,29)(22,36,38,39)(24,40,41,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388259);