Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C22×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2);(3,4);(9,10,12,16,13,17,21,24)(11,15,18,23,19,14,20,22);(5,6,7,8)(10,14,17,15)(16,23,24,22)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#13378, C2×#149485, C2×#147497, … |
| maximal normal subgroup of | C22×#13378, C2×#149485, C2×#147497, … |
| maximal quotient of | C22×#7308, #72267541, C2×#128060 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400098);