Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11393 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #379 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2710 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);(11,12,16,20,17,21,15,28)(13,22,24,18,25,33,29,32)(14,19,26,35,27,36,30,34)(23,38,41,42,40,37,39,31);(3,4,6,8)(5,9,10,7)(11,13)(12,18)(14,23)(15,29)(16,24)(17,25)(19,31)(20,33)(21,32)(22,28)(26,41)(27,40)(30,39)(34,37)(35,38)(36,42);(3,5,6,10)(4,7,8,9)(11,14)(12,19)(13,23)(15,30)(16,26)(17,27)(18,31)(20,35)(21,36)(22,37)(24,41)(25,40)(28,34)(29,39)(32,42)(33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265991);