Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 10787 conjugacy classes of subgroups; only the 3337 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,25,23)(24,26,28,27);(13,15)(14,17)(16,19)(18,20)(24,28)(26,27);(1,2,4,3)(6,9)(7,12)(14,18)(15,19)(21,22)(23,25)(24,26)(27,28);(2,3)(7,12)(10,11)(22,23)(26,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23,25,22)(24,26,28,27);(5,7,8,12)(6,10,9,11)(13,15)(14,17)(16,19)(18,20)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^10|a^2=h,c^2=gh,e^2=j,f^2=hi,c^a=cg,c^b=ci,d^a=dj,d^c=dj,e^a=ehj,e^c=eh,e^d=ej,f^a=fgij,f^b=fghj,f^c=fgj,f^d=fgij,f^e=fghj130 conjugacy classes — view the full character table.
| maximal subgroup of | #72232710, C2×#72232681, #72327354, … |
| maximal normal subgroup of | #72232710, C2×#72232681, #72327354, … |
| maximal quotient of | #72293988, C2×#72232681 |
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.8);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);