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×#51850 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4562 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)(11,12,16,19)(13,14,17,21)(15,18,23,25)(20,22,24,26)(28,32)(30,38)(33,35)(36,42);(13,17)(14,21)(15,23)(18,25)(27,28,31,32)(29,33,37,35)(30,39,38,34)(36,41,42,40);(3,4,6,7)(5,8,9,10)(11,13)(12,14)(15,20)(16,17)(18,22)(19,21)(23,24)(25,26)(27,28)(29,35)(30,34)(31,32)(33,37)(36,41)(38,39)(40,42);(3,5,6,9)(4,8,7,10)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,29,31,37)(28,33,32,35)(30,36,38,42)(34,40,39,41);(11,15,16,23)(12,18,19,25)(13,20,17,24)(14,22,21,26)(27,30,31,38)(28,34,32,39)(29,36,37,42)(33,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614970);