Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11837 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1514 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,13,15,8)(6,9)(7,12)(10,14,17,18)(11,16);(4,8)(5,14)(6,16)(9,17)(10,11)(12,13)(15,18);(4,9,12,11)(5,15)(8,17,13,10)(14,18)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#482709, #72250336, #72250372, … |
| maximal normal subgroup of | C2×#482709, #72250336, #72250372, … |
| maximal quotient of | #72240792, #72307327, #72307346 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389140);