Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1333 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | #2215 | C8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 316 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) = 96 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2,4,8,5,9,14,20,6,10,15,21,16,22,26,30)(3,7,12,18,13,19,25,29,11,17,23,27,24,28,31,32)(33,34)(35,46,48,75,50,76,47,74)(36,42)(37,54,56,87)(38,59)(39,63)(40,45)(41,49,64,78,65,82,67,79)(43,57,70,90)(44,61)(51,85,80,77,84,69,60,94)(52,86,58,73)(53,72)(55,71,88,66,89,96,83,92)(62,93)(68,81,95,91);(1,3)(2,7)(4,12)(5,13)(6,11)(8,18)(9,19)(10,17)(14,25)(15,23)(16,24)(20,29)(21,27)(22,28)(26,31)(30,32)(33,35)(34,41)(36,48)(37,55)(38,60)(39,47)(40,50)(42,67)(43,69)(44,71)(45,65)(46,73)(49,81)(51,53)(52,83)(54,75)(56,89)(57,78)(58,88)(59,92)(61,84)(62,80)(63,64)(66,93)(68,94)(70,85)(72,96)(74,87)(76,86)(77,95)(79,90)(82,91);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32)(33,36,40,39)(34,42,45,63)(35,47,50,48)(37,52,56,58)(38,53,62,61)(41,64,65,67)(43,68,70,95)(44,59,72,93)(46,74,76,75)(49,78,82,79)(51,60,84,80)(54,86,87,73)(55,88,89,83)(57,81,90,91)(66,96,92,71)(69,77,85,94)Polycyclic presentation:
2^10|a^2=d,c^2=j,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,b^a=be,c^b=chj,d^b=dij,e^a=eg,e^b=egij,e^d=ej,f^b=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72322405, #72322430, #72322475, … |
| maximal normal subgroup of | #72322405, #72322430, #72322475, … |
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, 1, F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*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);