Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| prime radical | C2×#15257 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 7110 conjugacy classes of subgroups; only the 1163 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,8)(10,12)(14,18)(15,20)(21,22,25,23)(24,26,28,27);(9,10)(11,12)(13,14)(15,19)(16,20)(17,18);(5,6)(7,8)(14,18)(15,20)(21,22,25,23)(24,27,28,26);(2,4)(5,6,7,8)(13,15,17,20)(14,19,18,16)(21,23,25,22)(24,27,28,26);(1,2)(3,4)(5,6)(7,8)(13,16,17,19)(14,15,18,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^10|a^2=i,c^2=i,d^2=gi,e^2=f,b^a=bj,c^b=cf,d^a=dg,d^c=dg,e^b=ef,e^c=ei,e^d=eh154 conjugacy classes — view the full character table.
| maximal subgroup of | #72333218, #72333263, #72333266, … |
| maximal normal subgroup of | #72333218, #72333263, #72333266, … |
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.7*F.9);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);