Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,32,33)(29,31,37,34)(30,38,39,35)(36,42,40,41);(3,4,7,9)(5,6,10,8)(12,16)(13,19)(14,23)(24,26)(28,33)(30,39)(31,34)(36,40);(3,5,7,10)(4,8,9,6)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29)(28,34)(30,40)(31,33)(32,37)(35,41)(36,39)(38,42);(3,5)(4,6)(7,10)(8,9)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,26)(23,25)(27,30,32,39)(28,35,33,38)(29,36,37,40)(31,41,34,42);(3,6)(4,5)(7,8)(9,10)(11,14,15,23)(12,18,16,22)(13,21,19,25)(17,24,20,26)(27,31,32,34)(28,29,33,37)(30,41,39,42)(35,36,38,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008176);