Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#51528 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(24,25)(27,35)(29,31)(33,38);(8,12)(9,15)(10,18)(20,22)(23,24,28,25)(26,31,34,29)(27,30,35,32)(33,37,38,36);(3,4)(5,6)(7,8)(9,16)(10,14)(11,12)(13,15)(17,22)(18,19)(20,21)(23,25,28,24)(26,31,34,29)(27,32,35,30)(33,37,38,36);(7,9)(8,13)(10,17)(11,15)(12,16)(14,20)(18,21)(19,22)(23,26)(24,29)(25,31)(27,33)(28,34)(30,36)(32,37)(35,38);(7,10,11,18)(8,14,12,19)(9,17,15,21)(13,20,16,22)(23,27)(24,30)(25,32)(26,33)(28,35)(29,36)(31,37)(34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614707);