Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #15259 |
| commutator, Frattini, cosocle | C26 | C24 |
| prime radical | #62817 | C22 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
3795 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 3795 conjugacy classes of subgroups — too many to draw. Only its 217 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) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,13,16,14)(12,15)(17,18)(19,23)(20,30)(21,34,36,25)(22,26)(24,33)(27,43)(28,38,35,41)(29,39)(31,37)(32,40,46,42)(44,48,45,47);(1,2)(3,9,11,6)(4,7,12,15)(5,8)(10,13)(14,16)(17,19,22,29)(18,23,26,39)(20,31)(21,35,36,28)(24,37)(25,41,34,38)(27,33)(30,43)(32,45)(40,47)(42,48)(44,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,16)(13,15)(17,20,22,33)(18,24)(19,27)(21,32)(23,37,39,43)(25,42,34,40)(26,30)(28,45,35,44)(29,31)(36,46)(38,47)(41,48);(1,4,5,12)(2,7,8,15)(3,10,11,16)(6,13,9,14)(17,21,22,36)(18,25,26,34)(19,28,29,35)(20,32,33,46)(23,38,39,41)(24,40,30,42)(27,44,31,45)(37,47,43,48)Polycyclic presentation:
2^10|a^2=g,b^2=hij,c^2=ef,d^2=i,b^a=bgh,c^a=cf,c^b=ce,d^a=defgij,d^b=dfgi,d^c=de,e^a=ei,e^b=ei,e^c=eij,f^a=fj,f^b=fj,f^c=fij,f^d=fij,g^c=gj,g^d=gij,h^c=hi,h^d=hj70 conjugacy classes — view the full character table.
| maximal subgroup of | #72249477, #72249481, #72249487, … |
| maximal normal subgroup of | #72249477, #72249481, #72249487, … |
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.8*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.6);;
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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);