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×#50432 |
| 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);(4,7)(8,10)(11,12)(13,14)(15,23)(16,17)(18,21)(19,24)(20,25)(22,26)(27,28,31,32)(29,33,35,39)(30,34,38,40)(36,41,37,42);(12,17)(14,18)(19,23)(22,25)(28,32)(29,35)(34,40)(36,37);(3,4,6,7)(5,8,9,10)(11,13)(12,18)(14,17)(15,20)(16,21)(19,25)(22,23)(24,26)(27,29)(28,33)(30,37)(31,35)(32,39)(34,42)(36,38)(40,41);(11,14,16,18)(12,13,17,21)(15,22,24,25)(19,20,23,26)(27,30,31,38)(28,34,32,40)(29,36,35,37)(33,41,39,42);(3,5,6,9)(4,8,7,10)(11,15,16,24)(12,19,17,23)(13,20,21,26)(14,22,18,25)(27,28)(29,33)(30,34)(31,32)(35,39)(36,41)(37,42)(38,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612999);