Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #6340119 |
| commutator, Frattini, cosocle | C25 | C25 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3537 conjugacy classes of subgroups; only the 461 normal subgroups are shown.
This group has 3537 conjugacy classes of subgroups — too many to draw. Only its 461 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) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,8,16,22)(5,17)(6,10)(7,14)(9,18)(12,21,26,19)(13,24)(15,23,29,30)(20,27)(25,31,28,32);(1,3)(2,7,10,11)(4,12)(5,13)(6,14)(8,21,22,19)(9,24,17,20)(15,28)(16,26)(18,27)(23,32,30,31)(25,29);(5,18)(8,22)(9,17)(15,29)(19,21)(31,32);(1,4)(2,8,10,22)(3,12)(5,15)(6,16)(7,19,11,21)(9,23,17,30)(13,25)(14,26)(18,29)(20,31,24,32)(27,28);(1,5)(2,9)(3,13)(4,15)(6,18)(7,20)(8,23)(10,17)(11,24)(12,25)(14,27)(16,29)(19,31)(21,32)(22,30)(26,28)Polycyclic presentation:
2^10|a^2=g,b^2=i,d^2=i,b^a=bgh,c^a=cg,c^b=cf,d^a=dfgi,d^b=dfi,d^c=di,e^a=eghj,e^b=egi,e^c=ehj,f^e=fj,g^d=gj,h^b=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72318468, #72319379, #72335106, … |
| maximal normal subgroup of | #72318468, #72319379, #72335106, … |
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.7);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);