Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41921 |
| 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 | |
4616 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)(11,12)(13,19)(14,20)(15,18)(16,17)(21,25)(22,26)(23,24)(27,28,32,31)(29,35,37,33)(30,38,40,34)(36,41,39,42);(3,4,7,6)(5,8,10,9)(12,17)(13,20)(18,24)(21,26)(28,31)(33,35)(34,38)(41,42);(3,5,7,10)(4,8,6,9)(11,13)(12,14)(15,21)(16,20)(17,19)(18,22)(23,26)(24,25)(27,29)(28,33)(30,39)(31,35)(32,37)(34,42)(36,40)(38,41);(3,4,7,6)(5,8,10,9)(11,14)(12,13)(15,22)(16,19)(17,20)(18,21)(23,25)(24,26)(27,30)(28,34)(29,36)(31,38)(32,40)(33,41)(35,42)(37,39);(3,6,7,4)(5,9,10,8)(11,15,16,23)(12,18,17,24)(13,21,20,26)(14,22,19,25)(27,31,32,28)(29,35,37,33)(30,38,40,34)(36,42,39,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547061);