Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46183 |
| 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 | |
4530 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);(3,4,7,6)(5,9,10,8)(12,16)(13,19)(18,23)(20,25)(28,32)(29,35)(30,37)(40,42);(12,16)(13,19)(14,21)(24,26)(27,28,31,32)(29,33,35,39)(30,38,37,34)(36,42,41,40);(3,5,7,10)(4,8,6,9)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,28)(29,33)(30,34)(31,32)(35,39)(36,40)(37,38)(41,42);(3,6,7,4)(5,9,10,8)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,29)(28,33)(30,36)(31,35)(32,39)(34,40)(37,41)(38,42);(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,30,31,37)(28,34,32,38)(29,36,35,41)(33,40,39,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609991);