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×#49446 |
| 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 | |
4600 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)(13,16)(14,17)(15,20)(18,23)(19,22)(21,25)(24,26)(27,28,32,35)(29,30,33,37)(31,34,39,41)(36,38,40,42);(3,5,7,6)(4,8,10,9)(12,13)(15,23)(17,19)(21,26)(29,33)(30,37)(31,39)(34,41);(3,4)(5,8)(6,9)(7,10)(11,13,16,12)(14,17,22,19)(15,20,23,18)(21,24,26,25)(27,29)(28,30)(31,36)(32,33)(34,38)(35,37)(39,40)(41,42);(3,5,7,6)(4,8,10,9)(11,14)(12,17)(13,19)(15,21)(16,22)(18,24)(20,25)(23,26)(27,30)(28,33)(29,35)(31,38)(32,37)(34,40)(36,41)(39,42);(3,6,7,5)(4,9,10,8)(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612123);