Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | #495677 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C2×C3×C8 |
| Frattini | C22×C8 | C22×A4 |
| Fitting | #495677 | C3 |
| socle | C2 | C8×#120⋊C3 |
| prime radical | #3025⋊C3 | C4 |
| cosocle | #3573 | C22×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C16 | |
395 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,8,4,9,13,24,7,12,16,27,19,30,37,48)(5,32,15,25,43,29,35,62,22,31,41,51,42,56,59,61,21,11,14,50,18,28,58,47,20,34,40,49,45,55,57,64,6,10,39,26,17,54,36,46,23,33,38,52,44,53,60,63);(1,23)(2,34)(3,41)(4,45)(5,22,20,6)(7,21)(8,52)(9,56)(10,33,31,11)(12,32)(13,60)(14,40,38,15)(16,39)(17,44,42,18)(19,43)(24,64)(25,51,49,26)(27,50)(28,55,53,29)(30,54)(35,59,57,36)(37,58)(46,63,61,47)(48,62)Polycyclic presentation:
2^2,3,2^7|b^2=d,d^2=e,e^2=j,f^2=j,g^2=j,f^a=fhi,f^c=g,g^a=gh,g^c=fgj,g^f=gj,h^c=ij,h^f=hj,i^c=hij,i^f=ij,i^g=ij128 conjugacy classes — view the full character table.
| maximal subgroup of | #72342027 |
| maximal normal subgroup of | #72342027 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);