Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50947 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4730 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);(4,7)(8,10)(11,12)(13,16)(14,17)(15,20)(18,23)(19,22)(21,25)(24,26)(27,28,32,36)(29,30,34,33)(31,35,39,42)(37,38,41,40);(12,13)(15,23)(17,19)(21,26)(29,34)(30,33)(31,39)(35,42);(3,4)(5,8)(6,7)(9,10)(11,13,16,12)(14,17,22,19)(15,20,23,18)(21,24,26,25)(27,29)(28,33)(30,36)(31,37)(32,34)(35,40)(38,42)(39,41);(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,30,32,33)(28,34,36,29)(31,38,39,40)(35,41,42,37);(3,5,6,9)(4,8,7,10)(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,39)(28,35,36,42)(29,37,34,41)(30,38,33,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613837);