Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37259 |
| 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 | |
6808 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)(9,15)(10,17)(11,19)(12,13)(14,20)(16,18)(21,22)(24,26)(29,31)(30,34)(36,37);(8,13)(10,15)(14,19)(18,21)(23,24,28,26)(25,31,33,29)(27,30,35,34)(32,37,38,36);(3,4)(5,6)(7,9)(8,10)(11,16)(12,17)(13,15)(14,18)(19,21)(20,22)(23,25,28,33)(24,29,26,31)(27,32,35,38)(30,36,34,37);(7,10)(8,9)(11,18)(12,15)(13,17)(14,16)(19,22)(20,21)(23,26,28,24)(25,31,33,29)(27,34,35,30)(32,37,38,36);(7,11,12,20)(8,14,13,19)(9,16,17,22)(10,18,15,21)(23,27,28,35)(24,30,26,34)(25,32,33,38)(29,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541820);