Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #2297 |
| commutator | C22×C4 | C22×C42 |
| Frattini, cosocle | C23×C8 | C24 |
| socle | C22 | #21379 |
| prime radical | C25 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,14)(10,13)(15,21)(16,23)(17,22)(18,24)(19,25)(20,26)(27,33)(28,32)(29,35)(30,34)(31,36);(5,7,8,16,9,15,19,27,10,17,18,30,20,28,31,29)(6,11,12,23,14,21,25,33,13,22,24,34,26,32,36,35);(7,15,17,28)(8,18)(11,21,22,32)(12,24)(16,29,30,27)(19,31)(23,35,34,33)(25,36);(11,22)(21,32)(23,34)(33,35)Polycyclic presentation:
2^10|a^2=e,b^2=f,c^2=h,f^2=i,i^2=j,c^b=cij,d^a=dh,d^b=dg,f^c=fj,g^a=gj,h^b=hj160 conjugacy classes — view the full character table.
| maximal subgroup of | #72239203, #72239434, #72249855, … |
| maximal normal subgroup of | #72239203, #72239434, #72249855, … |
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, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);