Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
At least 9782 conjugacy classes of subgroups; only the 3301 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(14,18)(16,19)(21,22)(23,24)(25,26)(27,28);(13,14)(15,19)(16,20)(17,18)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(6,9)(10,12)(14,18)(16,19)(22,26)(23,27);(1,2,3,4)(6,9)(7,11)(22,26)(24,28);(5,6)(7,12)(8,9)(10,11)(13,15)(14,19)(16,18)(17,20)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,20,18,15)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^10|d^2=i,f^2=hij,b^a=bg,c^a=cgh,c^b=chi,d^a=dgh,d^c=di,e^c=ej,e^d=ej,f^a=fgi,f^c=fg,f^d=fij,f^e=fhij124 conjugacy classes — view the full character table.
| maximal subgroup of | #72329738 |
| maximal normal subgroup of | #72329738 |
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, 4, F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);