Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#13180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
702 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);(3,4,6,5)(7,8)(9,19)(10,16)(11,26)(12,29)(13,18)(14,20)(15,24)(17,28)(21,34)(22,36)(23,30)(25,35)(27,31)(32,37)(33,38);(3,5,6,4)(7,9)(8,14)(10,20)(11,27)(12,21)(13,23)(15,30)(16,34)(17,31)(18,33)(19,26)(22,28)(24,37)(25,38)(29,32)(35,36);(7,10,12,24,13,25,28,11)(8,15,17,16,18,26,29,35)(9,20,21,37,23,38,22,27)(14,30,31,34,33,19,32,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402257);