Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56187 |
| 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 | |
4718 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);(3,4,8,6)(5,9,7,10)(11,12)(13,19)(14,21)(15,23)(16,17)(18,24)(20,22)(25,26)(27,28,32,35)(29,30,33,38)(31,34,40,42)(36,37,39,41);(4,6)(5,7)(12,17)(14,19)(15,24)(20,26)(27,29,32,33)(28,30,35,38)(31,39,40,36)(34,41,42,37);(3,5,8,7)(4,9,6,10)(11,13,16,21)(12,14,17,19)(15,20,24,26)(18,22,23,25)(27,29)(28,30)(31,36)(32,33)(34,37)(35,38)(39,40)(41,42);(3,6,8,4)(5,9,7,10)(11,14)(12,13)(15,22)(16,19)(17,21)(18,20)(23,26)(24,25)(27,30)(28,33)(29,35)(31,37)(32,38)(34,39)(36,42)(40,41);(3,7,8,5)(4,9,6,10)(11,15)(12,18)(13,20)(14,22)(16,24)(17,23)(19,25)(21,26)(27,31)(28,34)(29,36)(30,37)(32,40)(33,39)(35,42)(38,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009946);