Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C225 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2785720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 8 generators on 24 points:
(2,5)(6,8)(10,13)(14,16)(18,21)(22,24);(3,7)(6,8)(11,15)(14,16)(19,23)(22,24);(18,21)(22,24);(19,23)(22,24);(10,13)(14,16);(11,15)(14,16);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,18,20,21)(19,22,23,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^11|g^2=k,g^a=gk,g^c=gj,g^e=gi,h^b=hk,h^d=hj,h^f=hi368 conjugacy classes — view the full character table.
| Fitting subgroup of | #72255786, #72256075 |
| maximal subgroup of | C2×#72236376, #72255786, #72255989, … |
| maximal normal subgroup of | C2×#72236376, #72255786, #72255989, … |
| maximal quotient of | C2×#72236376 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);