Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C27 |
| direct product | C22×D4⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C24 | C22⋊C8 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6)(7,9)(8,11)(10,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(6,9)(7,12)(10,11)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#12062, C2×#483602, C2×#481074, … |
| maximal normal subgroup of | C22×#12062, C2×#483602, C2×#481074, … |
| maximal quotient of | C22×#7070, #72320645 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388236);