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 | |
3827 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 3827 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)(3,7)(4,11)(5,8)(6,9)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,32,34,25)(22,26)(27,35)(28,40,33,36)(29,37)(30,41,45,38)(31,39)(42,48,44,46)(43,47);(1,3)(2,7)(4,6)(5,10,13,12)(8,15,16,14)(9,11)(17,19)(18,23)(20,27)(21,33,34,28)(22,29)(24,35)(25,40,32,36)(26,37)(30,42,45,44)(31,43)(38,46,41,48)(39,47);(1,4,6,3)(2,7,9,11)(5,12,13,10)(8,15,16,14)(17,20,22,31)(18,24,26,39)(19,27,29,43)(21,30)(23,35,37,47)(25,41)(28,44)(32,38)(33,42)(34,45)(36,46)(40,48);(1,5,6,13)(2,8,9,16)(3,10,4,12)(7,14,11,15)(17,21,22,34)(18,25,26,32)(19,28,29,33)(20,30,31,45)(23,36,37,40)(24,38,39,41)(27,42,43,44)(35,46,47,48)Polycyclic presentation:
2^10|a^2=g,b^2=gh,c^2=gij,d^2=ij,c^a=cgj,c^b=chj,d^a=de,d^b=df,d^c=defij,e^a=ei,e^b=ej,e^c=ej,f^a=fj,f^b=fij,f^c=fij,g^d=gi,h^d=hj70 conjugacy classes — view the full character table.
| maximal subgroup of | #72249452, #72249467, #72249477, … |
| maximal normal subgroup of | #72249452, #72249467, #72249477, … |
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*F.8);;
gap> SetPower(coll, 3, F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);