Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10336 |
| commutator, Frattini, cosocle | C22×#120 | C23 |
| prime radical | #245225 | C2 |
| LCS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
3209 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 3209 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 32 points:
(1,2,4,6)(3,9,10,13)(5,8,12,14)(7,15,16,11)(17,18,20,23)(19,25,26,22)(21,24,28,30)(27,31,32,29);(2,6)(7,16)(8,14)(9,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31);(1,3,5,11)(2,7,8,9)(4,10,12,15)(6,13,14,16)(25,31)(26,32)Polycyclic presentation:
2^10|a^2=d,c^2=i,b^a=bd,c^a=ce,c^b=cf,d^c=dg,e^a=eg,e^b=egh,e^d=ei,f^a=fhij,f^d=fj,g^a=gi,g^b=gj,h^a=hj,h^b=hj58 conjugacy classes — view the full character table.
| maximal subgroup of | #72266233, #72266326, #72266328 |
| maximal normal subgroup of | #72266233, #72266326, #72266328 |
| maximal quotient of | #72316537, #72316571 |
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.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
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.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);