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×C4 |
| direct product | C2×#13262 |
| 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 | |
1266 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);(4,6)(7,8,11,16,12,27,25,34,14,18,24,33,13,17,10,23)(9,19,20,30,28,35,38,32,22,36,37,31,21,29,26,15);(3,4,5,6)(7,9,14,22)(8,15,18,32)(10,20,25,37)(11,26,24,38)(12,21,13,28)(16,29,33,35)(17,30,27,31)(19,34,36,23);(7,10,13,24,14,25,12,11)(8,16,27,34,18,33,17,23)(9,20,28,38,22,37,21,26)(15,29,31,36,32,35,30,19)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402466);