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×#50309 |
| 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 | |
4660 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)(17,20)(18,23)(24,26)(27,28,32,30)(29,34,37,33)(31,38,40,35)(36,41,39,42);(12,16)(13,19)(14,22)(24,26)(27,29,32,37)(28,33,30,34)(31,39,40,36)(35,42,38,41);(3,5,7,10)(4,8,6,9)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29)(28,34)(30,33)(31,36)(32,37)(35,42)(38,41)(39,40);(3,6,7,4)(5,9,10,8)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,28)(29,34,37,33)(31,38,40,35)(36,42,39,41);(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,31,32,40)(28,35,30,38)(29,36,37,39)(33,41,34,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613377);