Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| prime radical | C25×D4 | C22 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
28531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(10,12)(14,17)(15,19)(21,22,25,27)(23,28,24,26);(6,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(6,8)(14,17)(18,20)(21,23,25,24)(22,26,27,28);(1,2,3,4)(5,6,7,8)(13,14)(15,20)(16,17)(18,19)(21,24,25,23)(22,26,27,28);(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,28)(25,27)Polycyclic presentation:
2^10|a^2=i,c^2=i,d^2=hij,b^a=bf,c^a=ci,c^b=cfj,d^a=dg,d^b=dfgh,d^c=dghj,e^a=efij,e^d=efij160 conjugacy classes — view the full character table.
| maximal subgroup of | #72333208, #72333219, #72333243, … |
| maximal normal subgroup of | #72333208, #72333219, #72333243, … |
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.9);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);