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×#48834 |
| 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 | |
7208 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)(7,8,12,10)(9,15,17,13)(11,18,19,14)(16,20,22,21)(23,24,28,30)(25,31,32,26)(27,29,35,36)(33,37,38,34);(8,10)(13,15)(14,18)(20,21)(25,32)(26,31)(27,35)(29,36);(3,4,5,6)(7,9)(8,13)(10,15)(11,16)(12,17)(14,20)(18,21)(19,22)(23,25,28,32)(24,26,30,31)(27,33,35,38)(29,34,36,37);(7,10,12,8)(9,15,17,13)(11,18,19,14)(16,21,22,20)(23,26)(24,25)(27,34)(28,31)(29,33)(30,32)(35,37)(36,38);(7,11,12,19)(8,14,10,18)(9,16,17,22)(13,20,15,21)(23,27)(24,29)(25,33)(26,34)(28,35)(30,36)(31,37)(32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612024);