Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10336 |
| commutator, Frattini, cosocle | C22×#127 | C23 |
| prime radical | #35527 | C22 |
| LCS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
1891 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 1891 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,7,9,14)(3,10,12,13)(4,5)(15,16)(17,18,22,26)(19,27)(20,24,32,39)(21,33,34,25)(23,35)(28,41)(29,37)(30,42)(31,40,46,38)(36,45)(43,47)(44,48);(1,2,6,9)(3,11,12,8)(4,7,5,14)(10,16,13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,43)(32,44)(33,42)(34,35)(38,41)(39,47)(40,48)(45,46);(1,3)(2,8)(4,13)(5,10)(6,12)(7,15)(9,11)(14,16)(17,20,22,32)(18,24,26,39)(19,28,30,44)(21,31,34,46)(23,36,37,47)(25,40,33,38)(27,41,42,48)(29,45,35,43)Polycyclic presentation:
2^10|a^2=hj,b^2=ij,c^2=j,b^a=bd,c^a=ce,c^b=cf,d^a=di,d^c=dg,e^a=eij,e^b=egh,e^d=ej,f^a=fh,f^d=fi,f^e=fij,g^a=gj,g^b=gi,h^b=hi,h^c=hij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72341241, #72341244 |
| maximal normal subgroup of | #72341241, #72341244 |
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.8*F.10);;
gap> SetPower(coll, 2, F.9*F.10);;
gap> SetPower(coll, 3, F.10);;
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, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);