Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11934 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
784 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);(4,7)(5,9)(8,10)(11,12,16,22,17,23,32,39,18,24,30,37,14,20,15,21)(13,25,28,36,26,38,27,40,29,41,31,19,33,35,42,34);(3,4,6,8)(5,10,9,7)(11,13)(12,19)(14,26)(15,31)(16,27)(17,33)(18,29)(20,34)(21,38)(22,25)(23,40)(24,36)(28,32)(30,42)(35,39)(37,41);(11,14,18,17)(12,20,24,23)(13,26,29,33)(15,30,32,16)(19,34,36,40)(21,37,39,22)(25,38,41,35)(27,31,42,28)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390249);