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×#48775 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7440 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)(9,12)(10,13)(11,16)(14,19)(15,18)(17,21)(20,22)(23,24,28,31)(25,26,29,33)(27,30,35,37)(32,34,36,38);(8,9)(11,19)(13,15)(17,22)(25,29)(26,33)(27,35)(30,37);(3,4)(5,6)(7,9,12,8)(10,13,18,15)(11,16,19,14)(17,20,22,21)(23,25)(24,26)(27,32)(28,29)(30,34)(31,33)(35,36)(37,38);(7,10)(8,13)(9,15)(11,17)(12,18)(14,20)(16,21)(19,22)(23,26)(24,29)(25,31)(27,34)(28,33)(30,36)(32,37)(35,38);(7,11)(8,14)(9,16)(10,17)(12,19)(13,20)(15,21)(18,22)(23,27)(24,30)(25,32)(26,34)(28,35)(29,36)(31,37)(33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611407);