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 | |
1674 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 1674 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2)(3,7)(4,6)(5,12)(8,11)(9,15)(10,13)(14,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,43)(31,55)(33,65)(34,57)(35,67)(36,48)(37,49)(38,50)(39,52)(40,56)(41,53)(42,54)(44,72)(45,70)(46,58)(47,68)(51,59)(60,71)(61,79)(62,75)(63,74)(64,77)(66,78)(69,73)(76,80);(2,6)(3,9)(7,14)(8,12)(10,16)(13,15)(17,19,30,58)(18,24,43,57)(20,35,59,80)(21,32,23,34)(22,40,60,64)(25,47,51,78)(26,29,28,46)(27,45,71,65)(31,61,72,37)(33,39,70,42)(36,68,38,66)(41,63,69,62)(44,73,55,53)(48,67,50,76)(49,75,79,74)(52,56,54,77);(1,3)(2,7)(4,10)(5,9)(6,13)(8,15)(11,16)(12,14)(17,20,23,38)(18,25,28,50)(19,31,34,63)(21,36,30,59)(22,41,42,37)(24,44,46,75)(26,51,43,48)(27,53,54,49)(29,55,57,74)(32,62,58,72)(33,66,40,35)(39,61,60,69)(45,76,77,47)(52,73,71,79)(56,67,65,78)(64,80,70,68)Polycyclic presentation:
2^10|b^2=dij,c^2=j,e^2=j,b^a=bd,c^a=cf,c^b=ce,d^c=dh,e^a=eg,e^b=ehj,e^c=ej,f^b=fghij,f^d=fj,f^e=fi,g^b=gj,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.4*F.9*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.8*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, 2, F.7*F.10);;
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);