Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41623 |
| 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 | |
4534 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)(11,12,16,13)(14,19,22,17)(15,18,23,20)(21,25,26,24)(27,28)(29,35)(30,38)(31,32)(33,37)(34,40)(36,41)(39,42);(12,13)(17,19)(18,20)(24,25)(28,32)(33,35)(34,38)(41,42);(3,4)(5,8)(6,7)(9,10)(11,13,16,12)(14,19,22,17)(15,20,23,18)(21,25,26,24)(27,29,31,37)(28,33,32,35)(30,39,40,36)(34,42,38,41);(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,30,31,40)(28,34,32,38)(29,36,37,39)(33,41,35,42);(3,5,6,9)(4,8,7,10)(11,15,16,23)(12,18,13,20)(14,21,22,26)(17,24,19,25)(27,28)(29,33)(30,34)(31,32)(35,37)(36,41)(38,40)(39,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547302);