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×#50168 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6792 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);(4,6)(7,8,12,14)(9,15,16,10)(11,13,19,20)(17,21,22,18)(23,24,28,31)(25,32,27,29)(26,30,35,38)(33,37,34,36);(9,16)(10,15)(11,19)(13,20)(23,25,28,27)(24,29,31,32)(26,34,35,33)(30,37,38,36);(3,4)(5,6)(7,9,12,16)(8,10,14,15)(11,17,19,22)(13,18,20,21)(23,26,28,35)(24,30,31,38)(25,33,27,34)(29,36,32,37);(7,10)(8,9)(11,18)(12,15)(13,17)(14,16)(19,21)(20,22)(23,27,28,25)(24,29,31,32)(26,33,35,34)(30,37,38,36);(7,11)(8,13)(9,17)(10,18)(12,19)(14,20)(15,21)(16,22)(23,24)(25,32)(26,30)(27,29)(28,31)(33,37)(34,36)(35,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613034);