Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#1465 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
6418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,11,12,9)(8,10)(13,14,16,18,17,19,15,20);(5,7)(6,9)(8,12)(10,11);(13,15,17,16)(14,18,19,20)128 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#18166, C22×#17362, C2×#6346601 |
| maximal normal subgroup of | C22×#18166, C22×#17362, C2×#6346601 |
| maximal quotient of | C2×#207786, #72309364 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249748);