Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13384 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,5)(7,8)(9,13)(10,14)(11,16)(12,18)(15,23)(17,24)(19,32)(20,29)(21,27)(22,28)(25,31)(26,33)(30,36)(34,38)(35,37);(3,4)(5,6)(7,9)(8,13)(10,20)(11,19)(12,22)(14,29)(15,26)(16,32)(17,27)(18,28)(21,24)(23,33)(25,34)(30,37)(31,38)(35,36);(3,5,6,4)(8,14)(9,19)(10,23)(11,24)(13,26)(15,17)(16,30)(18,31)(20,34)(21,22)(25,36)(27,38)(28,37)(29,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406204);