Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(8,15)(9,21)(10,26)(11,29)(12,13)(16,33)(17,20)(18,27)(19,30)(23,31)(24,28)(25,37)(32,34)(36,38);(3,4)(5,6)(7,8,14,20)(9,22,25,35)(10,27,11,30)(12,19,13,18)(15,29,17,26)(16,24,33,23)(21,34,37,38)(28,36,31,32);(4,6)(7,9)(8,16)(10,28)(11,31)(12,24)(13,23)(14,25)(15,32)(17,36)(18,35)(19,22)(20,33)(21,29)(26,37)(27,38)(30,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406071);