Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1343 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#381 | C23 |
| prime radical | C22×#534 | C4 |
| LCS | G ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C26 ◁ C22×#381 ◁ G | |
| LECS | G ▷ C2×#381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
4776 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 4776 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,8,13)(7,15,16,31,17,30,20,34)(9,22,24,35)(10,11,19,26)(12,27,28,18,25,32,36,33)(14,21,29,23);(2,4)(6,11)(8,19)(9,23)(12,25)(16,20)(18,33)(29,35)(31,34);(5,7,10,17)(6,12,11,25)(8,20,19,16)(9,18,23,33)(13,28,26,36)(14,30,22,15)(21,27,24,32)(29,31,35,34)Polycyclic presentation:
2^10|a^2=d,c^2=j,d^2=h,f^2=hi,b^a=be,c^a=cf,d^b=dj,d^c=dg,e^a=ej,e^c=ej,f^a=fghi,f^b=fj,f^c=fhi,f^d=fij,g^a=gi,g^d=gj,g^f=gj,h^c=hj,i^a=ij,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72285209 |
| maximal normal subgroup of | #72285209 |
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.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);