Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #9041 |
| commutator | C2×#97 | C22×C4 |
| Frattini, cosocle | C22×#97 | C23 |
| prime radical | #499940 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 97 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 — 3 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(9,16)(10,13)(11,14)(12,15)(18,23)(19,27)(20,31)(21,34)(24,37)(25,38)(26,41)(28,39)(29,45)(32,35)(33,36)(40,48)(42,47)(44,46);(1,3)(2,6)(4,9)(5,11)(7,14)(8,16)(10,12)(13,15)(17,18,22,25)(19,28,27,24)(20,32,33,41)(21,35,34,26)(23,36,38,31)(29,46,42,48)(30,40,43,44)(37,45,39,47);(2,7)(3,9,11,10)(6,13,14,16)(8,15)(17,19)(18,24,25,28)(20,29)(21,30)(22,27)(23,37,38,39)(26,40,35,44)(31,47)(32,46,41,48)(33,42)(34,43)(36,45)Polycyclic presentation:
2^10|b^2=j,c^2=f,d^2=h,e^2=ij,i^2=j,b^a=be,c^a=cghj,c^b=cd,d^a=dg,d^b=dh,d^c=dj,e^a=ei,e^b=ei,e^c=eg,e^d=ej,f^b=fhj,g^b=gj,i^a=ij,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72265103, #72265517, #72265552 |
| maximal normal subgroup of | #72265103, #72265517, #72265552 |
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, 2, F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
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.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);