Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C217 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C26 |
| commutator | C23 | C25×C4 |
| prime radical | C24×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
248917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(13,14,18,16)(15,19,21,23)(17,20,25,24)(22,26,27,28);(10,12)(14,16)(19,23)(20,24)(26,28);(2,4)(6,8)(10,12)(15,21)(19,23)(22,27)(26,28);(10,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27);(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(13,16,18,14)(15,19,21,23)(17,20,25,24)(22,28,27,26);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^10|a^2=j,e^2=g,b^a=bj,d^c=dj,e^b=eh,e^c=ei,e^d=eh,f^e=fj232 conjugacy classes — view the full character table.
| maximal subgroup of | #72337295, C2×#72337292 |
| maximal normal subgroup of | #72337295, C2×#72337292 |
| maximal quotient of | #72337316, #72337318, #72337391, … |
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.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);