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 | |
At least 9783 conjugacy classes of subgroups; only the 3301 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(15,19)(17,20)(22,26)(24,28);(6,7)(10,11)(21,22)(23,24)(25,26)(27,28);(2,4)(5,6,9,7)(8,11,12,10)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(6,7)(10,11)(13,14,16,18)(15,20,19,17)(21,23,25,27)(22,24,26,28);(5,7,9,6)(8,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^10|c^2=g,d^2=h,e^2=g,b^a=bi,c^b=cg,d^c=dgj,e^b=eg,e^d=egh,f^a=fh,f^c=fgij,f^d=fh124 conjugacy classes — view the full character table.
| maximal subgroup of | #72300575 |
| maximal normal subgroup of | #72300575 |
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, 3, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);