Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C16 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7136 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C8 | C22 |
| commutator | C2 | C2×C8×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C24 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2);(3,4)(5,6)(15,16,17,20,18,21,23,26,19,22,24,27,25,28,29,30);(4,6)(7,8,9,11,10,12,13,14)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#128204 |
| maximal normal subgroup of | C2×#128204 |
| maximal quotient of | C2×#93249 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33742);