Finite Group DatabasebetaFinite Group Databasebeta| order | 8192 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C213 |
| direct product | C2×#72216741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #72216737 |
| commutator, Frattini, cosocle | C2×#2846 | C25 |
| LCS | G ▷ C2×#2846 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C26 ◁ C22×#2846 ◁ G | |
| LECS | G ▷ C2×#2846 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
At least 53390 conjugacy classes of subgroups; only the 846 normal subgroups are shown.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(3,9,11,12)(6,14,10,16);(1,2,5,8)(3,10)(4,7,13,15)(6,11)(9,16)(12,14);(7,15)(9,12)(14,16);(1,3)(2,6)(4,12)(5,11)(7,16)(8,10)(9,13)(14,15);(17,18)212 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,9,11,12)(6,14,10,16), (1,2,5,8)(3,10)(4,7,13,15)(6,11)(9,16)(12,14), (7,15)(9,12)(14,16), (1,3)(2,6)(4,12)(5,11)(7,16)(8,10)(9,13)(14,15), (17,18));