Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 6807 conjugacy classes of subgroups; only the 4421 normal subgroups are shown.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(10,12)(13,14)(15,18)(16,17)(19,20);(6,7)(9,10)(11,12)(14,17)(15,19);(13,14)(15,18)(16,17)(19,20);(2,4)(5,6,8,7)(14,17)(15,19);(2,4)(5,7,8,6)(13,15,16,19)(14,18,17,20);(1,2)(3,4)(13,14)(15,20)(16,17)(18,19)Polycyclic presentation:
2^10|d^2=hij,e^2=hj,b^a=bh,c^b=ci,d^a=di,d^b=dhij,d^c=di,e^b=ehj,e^d=ei,f^a=fg,f^b=fi,f^d=fgi,f^e=fgi172 conjugacy classes — view the full character table.
| maximal subgroup of | #72298104, C2×#72298086 |
| maximal normal subgroup of | #72298104, C2×#72298086 |
| maximal quotient of | #72298589, C2×#72298086 |
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, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);