Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37564 |
| 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 | |
4486 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);(3,4,6,8)(5,9,10,7)(14,22)(17,24)(20,26)(23,25)(27,28,31,35)(29,33,37,32)(30,38,39,34)(36,40,42,41);(11,12,15,18)(13,16,21,19)(14,17,22,24)(20,23,26,25)(27,29)(28,32)(30,36)(31,37)(33,35)(34,40)(38,41)(39,42);(3,5,6,10)(4,7,8,9)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,29)(28,33)(30,36)(31,37)(32,35)(34,41)(38,40)(39,42);(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25)(27,28)(29,32)(30,34)(31,35)(33,37)(36,40)(38,39)(41,42);(11,14)(12,17)(13,20)(15,22)(16,23)(18,24)(19,25)(21,26)(27,30,31,39)(28,34,35,38)(29,36,37,42)(32,40,33,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541025);