Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2840 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | C23×#127 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
5057 conjugacy classes of subgroups; only the 497 normal subgroups are shown.
This group has 5057 conjugacy classes of subgroups — too many to draw. Only its 497 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7,12,15)(5,8)(6,11)(10,14,16,13)(17,18)(19,25)(20,28,30,23)(21,24)(22,26)(27,32,29,31);(1,3)(2,6)(4,10,12,16)(5,11)(7,14,15,13)(8,9)(17,19,21,26)(18,22,24,25)(20,29)(23,32)(27,30)(28,31);(19,26)(22,25)(27,29)(31,32);(6,9)(13,14)(17,20,21,30)(18,23,24,28)(19,27,26,29)(22,31,25,32);(1,4,5,12)(2,7,8,15)(3,10,11,16)(6,13,9,14)(18,24)(23,28)Polycyclic presentation:
2^10|a^2=f,b^2=fj,d^2=j,e^2=i,b^a=bfij,c^b=cj,d^a=dh,d^b=dg,e^a=egij,e^b=eghi,f^d=fj,f^e=fi,g^a=gi,g^b=gj,h^a=hj,h^b=hij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72338964, #72338981, #72338992, … |
| maximal normal subgroup of | #72338964, #72338981, #72338992, … |
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.6);;
gap> SetPower(coll, 2, F.6*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);