Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10336 |
| commutator, Frattini, cosocle | C22×#123 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1 | |
2349 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 2349 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(5,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,51)(24,49)(25,50)(26,57)(27,39)(28,40)(29,41)(30,43)(31,44)(32,64)(33,46)(34,47)(35,52)(36,54)(37,55)(38,58)(42,60)(45,61)(48,53)(56,68)(59,67)(62,69)(63,70)(65,71)(66,72);(2,5)(6,8)(9,11,15,25)(10,16,20,37)(12,26,29,56)(13,23,31,55)(14,32,34,24)(17,38,41,66)(18,21,44,50)(19,45,47,36)(22,33,53,28)(27,58,60,72)(30,61,63,54)(35,46,65,40)(39,57,42,68)(43,64,70,49)(48,67,51,69)(52,62,71,59);(1,2)(3,5)(4,6)(7,8)(9,12,15,29)(10,17,20,41)(11,22,25,53)(13,27,31,60)(14,33,34,28)(16,35,37,65)(18,42,44,39)(19,46,47,40)(21,48,50,51)(23,52,55,71)(24,56,32,26)(30,62,63,59)(36,66,45,38)(43,67,70,69)(49,57,64,68)(54,72,61,58)Polycyclic presentation:
2^10|b^2=j,c^2=j,e^2=j,b^a=bd,c^a=cf,c^b=ce,d^c=dh,e^a=eg,e^b=ej,e^c=ej,f^b=fghij,f^d=fj,f^e=fi,g^c=gi,h^a=hj49 conjugacy classes — view the full character table.
| maximal subgroup of | #72227725 |
| maximal normal subgroup of | #72227725 |
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.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);