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×#50880 |
| 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 | |
4686 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);(3,4,7,5)(6,9,10,8)(11,12)(13,16)(14,17)(15,20)(18,23)(19,22)(21,25)(24,26)(27,28,32,35)(29,33,36,41)(30,31,34,40)(37,42,39,38);(3,5,7,4)(6,9,10,8)(12,13)(15,23)(17,19)(21,26)(29,36)(30,34)(31,40)(33,41);(3,6)(4,8)(5,9)(7,10)(11,13,16,12)(14,17,22,19)(15,20,23,18)(21,24,26,25)(27,29)(28,33)(30,39)(31,38)(32,36)(34,37)(35,41)(40,42);(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,30,32,34)(28,31,35,40)(29,37,36,39)(33,42,41,38);(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)(27,31)(28,34)(29,38)(30,35)(32,40)(33,37)(36,42)(39,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613817);