Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12687 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | #2297 | C23 |
| prime radical | C2×H(7,𝔽2) | C4 |
| LCS | G ▷ C22×#41 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #35616 ◁ G | |
| LECS | G ▷ #2297 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
8697 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 8697 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(9,10)(11,23)(12,29)(13,20)(14,37)(15,19)(16,22)(17,26)(18,32)(21,40)(24,56)(25,52)(27,43)(28,55)(30,41)(31,66)(33,53)(34,42)(35,67)(36,48)(38,49)(39,47)(44,61)(45,59)(46,68)(50,64)(51,69)(54,58)(57,70)(60,72)(62,71)(63,65);(1,2,3,6,4,5,7,8)(9,11)(10,17,19,42)(12,30)(13,34)(14,25)(15,27,40,64)(16,28)(18,45,29,65)(20,50,48,55)(21,52,22,44)(23,37,43,49)(24,31)(26,38,61,36)(32,57,39,62)(33,60)(35,59)(41,66,70,46)(47,56,53,54)(51,72,67,71)(58,69,63,68);(2,5)(3,7)(9,12,16,33)(10,18,22,47)(11,24,28,59)(13,35,14,31)(15,39,40,32)(17,41,44,72)(19,29,21,53)(20,51,37,46)(23,54,55,65)(25,60,34,30)(26,62,61,57)(27,63,64,58)(36,68,38,69)(42,70,52,71)(43,56,50,45)(48,67,49,66)Polycyclic presentation:
2^10|b^2=d,c^2=j,d^2=h,e^2=j,f^2=j,g^2=j,b^a=bf,c^a=cg,c^b=ce,d^a=dg,d^c=dh,e^b=ehj,e^c=ej,f^a=fj,f^b=fg,f^c=fij,f^d=fij,g^a=gj,g^b=gi,g^c=gj,g^d=gj,g^f=gj,i^a=ij,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72228042 |
| maximal normal subgroup of | #72228042 |
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.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*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.7);;
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, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, 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);