Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #7103 |
| commutator | C22×#95 | C2×C8 |
| Frattini, cosocle | #94849 | C22 |
| prime radical | C26 | #99 |
| LCS | G ▷ C22×#95 ▷ C24×C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ C25×C4 ◁ #94849 ◁ G | |
| LECS | G ▷ #94849 ▷ C25×C4 ▷ C24 ▷ C22 ▷ C1 | |
3621 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 3621 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 2 generators on 48 points:
(1,2)(3,10,11,16,13,15,12,6)(4,8)(5,9)(7,14)(17,18,20,23)(19,22,25,28,26,29,32,30)(21,24,27,31)(33,34)(35,41)(36,39,45,47)(37,40)(38,42)(43,48)(44,46);(1,3,4,11,5,13,14,12)(2,6,8,10,9,16,7,15)(17,19,21,26)(18,22)(20,25)(23,30,31,28)(24,29)(27,32)(33,35)(34,38)(36,44,45,42)(37,43)(39,41,47,48)(40,46)Polycyclic presentation:
2^11|a^2=c,b^2=efj,c^2=f,d^2=gij,e^2=ij,b^a=bd,c^b=ce,d^a=deg,d^b=dghij,d^c=dhk,e^a=eh,e^b=ei,e^c=ej,e^d=ek,f^b=fi,g^a=gijk,h^a=hj,h^b=hk110 conjugacy classes — view the full character table.
| maximal subgroup of | #72267867 |
| maximal normal subgroup of | #72267867 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.5*F.6*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);