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×#50868 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,5,6)(8,12)(13,16)(14,19)(20,22)(23,24)(25,28)(26,29)(27,32)(30,35)(31,34)(33,37)(36,38);(8,12)(9,15)(10,18)(20,22)(24,25)(27,35)(29,31)(33,38);(3,4)(5,6)(7,8)(9,16)(10,14)(11,12)(13,15)(17,22)(18,19)(20,21)(23,25,28,24)(26,29,34,31)(27,32,35,30)(33,36,38,37);(7,9,11,15)(8,13,12,16)(10,17,18,21)(14,20,19,22)(23,26,28,34)(24,29,25,31)(27,33,35,38)(30,36,32,37);(7,10)(8,14)(9,17)(11,18)(12,19)(13,20)(15,21)(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, 7613767);