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 | |
32454 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)(7,10)(8,12)(9,11)(13,14)(15,16)(17,19)(18,20)(21,22,24,26)(23,25,27,28);(5,7)(6,10)(8,11)(9,12)(15,20)(16,18)(23,27)(25,28);(2,4)(7,12)(8,10)(13,15,17,20)(14,18,19,16)(23,27)(25,28);(2,4)(21,22,24,26)(23,28,27,25);(2,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,8,9,10)(6,7,11,12)(13,16,17,18)(14,15,19,20)(21,23,24,27)(22,25,26,28)Polycyclic presentation:
2^10|a^2=h,c^2=g,d^2=h,f^2=hi,c^a=cg,c^b=ci,e^a=eh,e^d=eh,f^a=fgi,f^b=fgh,f^c=fij,f^d=fj,f^e=fhij130 conjugacy classes — view the full character table.
| maximal subgroup of | #72232710, #72233450, #72233502, … |
| maximal normal subgroup of | #72232710, #72233450, #72233502, … |
| maximal quotient of | #72293994, C2×#72232670 |
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.8);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);