Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
32457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(5,6)(7,9)(8,11)(10,12)(15,19)(16,20)(21,22)(23,25)(24,27)(26,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,19)(17,18)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,17,19)(14,16,18,20);(1,2,3,4)(7,12)(9,10)(14,18)(15,19)(21,22)(23,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,25)(24,27)(26,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,18)(17,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^10|c^2=h,d^2=j,c^a=ch,c^b=chi,d^b=dg,d^c=dhj,e^d=ehj,f^a=fg,f^d=fij,f^e=fg136 conjugacy classes — view the full character table.
| maximal subgroup of | #72232676, #72233139, #72233191, … |
| maximal normal subgroup of | #72232676, #72233139, #72233191, … |
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.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);