Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C26 |
| commutator | C23 | C25×C4 |
| prime radical | C24×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
67669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(17,18,21,22)(19,23,24,20);(1,2,3,4)(6,8)(10,13)(11,15)(19,24)(20,23);(2,4)(5,6,7,8)(11,15)(14,16)(17,19,21,24)(18,20,22,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(1,2)(3,4)(5,6)(7,8);(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^10|a^2=i,b^2=g,c^2=gh,c^a=ci,c^b=ch,d^b=dhij,d^c=dhi,e^b=ehi,e^c=ehi,f^b=fij,f^c=fij208 conjugacy classes — view the full character table.
| maximal subgroup of | #72337332, #72339908, C2×#72337327 |
| maximal normal subgroup of | #72337332, #72339908, C2×#72337327 |
| maximal quotient of | #72337347, #72337349, #72337374, … |
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, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);