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×#55981 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4794 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);(3,4,8,9)(5,7,6,10)(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(28,29)(30,37)(31,39)(40,41);(4,9)(5,6)(13,20)(16,19)(21,26)(23,25)(27,28,32,29)(30,35,37,33)(31,36,39,34)(38,40,42,41);(3,5,8,6)(4,10,9,7)(11,13)(12,16)(14,21)(15,20)(17,23)(18,19)(22,26)(24,25)(27,29,32,28)(30,33,37,35)(31,36,39,34)(38,40,42,41);(3,6,8,5)(4,7,9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,30,32,37)(28,33,29,35)(31,38,39,42)(34,40,36,41);(3,7)(4,5)(6,9)(8,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009973);