Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23900 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15)(4,12)(5,13)(6,8)(7,14)(9,21)(10,23)(11,17)(16,26)(18,28)(19,27)(20,31)(22,30)(24,25)(29,32);(2,8)(3,16)(9,26)(10,14)(11,12)(13,25)(15,27)(18,32)(19,21)(20,22)(28,30)(29,31);(1,3,17,19)(2,9,11,26)(4,16,6,21)(5,18,7,22)(8,15,12,27)(10,30,25,28)(13,31,14,29)(20,23,32,24);(2,10)(3,18,32,16)(4,17)(5,24)(8,14)(9,26,30,28)(11,13)(12,25)(15,27,29,31)(19,21,20,22)Polycyclic presentation:
2^10|c^2=fh,d^2=fghij,e^2=j,b^a=bh,c^a=ch,c^b=cf,d^a=dfg,d^b=dfghij,d^c=de,e^a=eh,e^b=ei,e^c=ehij,e^d=eh,f^d=fhi,f^e=fj,g^c=gh,g^d=ghi,g^e=gj,i^c=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72240176, #72262846, #72262858, … |
| maximal normal subgroup of | #72240176, #72262846, #72262858, … |
| maximal quotient of | #72251826, #72318813, #72318854, … |
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, 3, F.6*F.8);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
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.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);