Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 11308 conjugacy classes of subgroups; only the 3353 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(13,14)(15,18)(16,20)(17,19)(22,24)(26,27);(2,4)(7,11)(9,12)(21,22,25,24)(23,27,28,26);(5,6,8,10)(7,9,11,12)(13,15,17,20)(14,18,19,16)(21,23)(22,26)(24,27)(25,28);(7,11)(9,12)(15,20)(16,18)(21,22,25,24)(23,27,28,26);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,16)(14,15)(17,18)(19,20)(21,24,25,22)(23,27,28,26);(5,6)(7,12)(8,10)(9,11)(13,14)(15,16)(17,19)(18,20)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^10|b^2=j,c^2=h,d^2=j,e^2=gh,b^a=bj,c^b=cj,d^a=dj,d^c=dg,e^a=eg,e^b=ei,e^d=eh,f^c=fh,f^e=fgh130 conjugacy classes — view the full character table.
| maximal subgroup of | #72232438, #72232573, C2×#72231231, … |
| maximal normal subgroup of | #72232438, #72232573, C2×#72231231, … |
| maximal quotient of | #72278177, #72278574, #72293994, … |
| automorphism group of | #1291 |
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.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);