Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8534 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×#120 | C23 |
| prime radical | #15258 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
2203 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 2203 conjugacy classes of subgroups — too many to draw. Only its 93 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 — 3 generators on 32 points:
(1,2,4,10)(3,13,14,26)(5,11,18,8)(6,22,21,25)(7,12,20,24)(9,19,23,17)(15,27,31,29)(16,28,30,32);(2,8)(4,18)(9,26)(10,11)(12,25)(13,23)(14,30)(19,31)(20,21)(22,24)(27,32)(28,29);(1,3,5,16)(2,9,22,13)(4,19,18,31)(6,17,7,15)(8,23,24,26)(10,29,25,32)(11,27,12,28)(14,20,30,21)Polycyclic presentation:
2^10|a^2=d,c^2=hj,b^a=bde,c^a=cf,c^b=cghj,d^b=dg,d^c=dh,e^a=egi,e^b=egi,e^c=ej,e^d=ei,f^a=fh,f^b=fhij,f^c=fj,f^d=fj,g^a=gi,h^a=hj58 conjugacy classes — view the full character table.
| maximal subgroup of | #72317073, #72320656, #72320678 |
| maximal normal subgroup of | #72317073, #72320656, #72320678 |
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.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);