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×#51298 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4730 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)(27,28,32,30)(29,33,36,37)(31,34,40,38)(35,41,39,42);(12,13)(15,23)(17,19)(21,26)(27,29,32,36)(28,33,30,37)(31,39,40,35)(34,42,38,41);(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)(28,33)(30,37)(31,35)(32,36)(34,41)(38,42)(39,40);(3,5,7,6)(4,8,10,9)(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,30,32,28)(29,33,36,37)(31,38,40,34)(35,41,39,42);(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)(29,35)(30,38)(32,40)(33,41)(36,39)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614172);