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×#41850 |
| 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);(3,4,7,10)(5,8,6,9)(11,12,16,17)(13,19,21,15)(14,22,25,18)(20,24,23,26)(27,28,32,36)(29,33,37,42)(30,34,40,39)(31,35,41,38);(3,5,7,6)(4,8,10,9)(12,17)(15,19)(18,22)(24,26)(29,37)(31,41)(33,42)(35,38);(3,4)(5,8)(6,9)(7,10)(11,13)(12,15)(14,23)(16,21)(17,19)(18,26)(20,25)(22,24)(27,29)(28,33)(30,35)(31,39)(32,37)(34,41)(36,42)(38,40);(3,5,7,6)(4,8,10,9)(11,14,16,25)(12,18,17,22)(13,20,21,23)(15,24,19,26)(27,30)(28,34)(29,38)(31,33)(32,40)(35,37)(36,39)(41,42);(3,6,7,5)(4,9,10,8)(11,15)(12,13)(14,24)(16,19)(17,21)(18,20)(22,23)(25,26)(27,31)(28,35)(29,39)(30,33)(32,41)(34,37)(36,38)(40,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547185);