Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23984 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C24 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
3405 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 3405 conjugacy classes of subgroups — too many to draw. Only its 141 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 — 4 generators on 32 points:
(1,2)(3,13,17,25)(4,11)(5,10)(6,8)(7,12)(9,16,23,20)(14,26,32,27)(15,30,18,31)(19,29)(21,28)(22,24);(2,8)(3,14,17,32)(4,19)(9,25,23,13)(10,12)(11,28)(15,16,18,20)(21,22)(24,29)(26,30,27,31);(1,3)(2,9)(4,20)(5,15)(6,17)(7,18)(8,23)(10,27)(11,25)(12,26)(13,24)(14,19)(16,22)(21,32)(28,31)(29,30);(2,10)(3,15)(4,21)(8,12)(13,30)(17,18)(19,22)(25,31)Polycyclic presentation:
2^10|a^2=g,b^2=g,e^2=i,b^a=bghij,c^a=ce,c^b=cef,d^a=dghi,d^b=dghj,d^c=dij,e^b=eh,e^c=ei,e^d=eij,f^a=fhi,f^b=fhj,f^c=fj,f^e=fj,g^c=gi,h^c=hj58 conjugacy classes — view the full character table.
| maximal subgroup of | #72233377, #72233463, #72233500, … |
| maximal normal subgroup of | #72233377, #72233463, #72233500, … |
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.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);