Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#1368 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #106 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#106 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4);(5,6,7,9,8,10,11,12)(14,18,19,24)(15,21)(16,23)(20,27,26,25)(22,28);(13,14,15,18,16,20,21,25,17,19,22,24,23,26,28,27)128 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12067, #72223570, C8×#1368, … |
| maximal normal subgroup of | C4×#12067, #72223570, C8×#1368, … |
| maximal quotient of | C8×#1368, C2×C4×#1368, C4×#7457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36285);