Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37090 |
| 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 | |
4562 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)(12,16)(14,21)(17,22)(20,25)(27,28,33,36)(29,31,30,34)(32,40,41,35)(37,42,38,39);(12,16)(13,19)(18,23)(20,25)(27,29,33,30)(28,34,36,31)(32,37,41,38)(35,42,40,39);(3,4,6,7)(5,8,9,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,31,33,34)(28,29,36,30)(32,39,41,42)(35,37,40,38);(3,5,6,9)(4,8,7,10)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,32)(28,35)(29,37)(30,38)(31,39)(33,41)(34,42)(36,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541549);