Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #499832 |
| commutator, Frattini, cosocle | C2×#127 | C24 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2743 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 2743 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,9)(5,8)(6,12)(7,11)(13,23)(14,24)(15,25)(16,30)(17,32)(18,26)(19,28)(20,27)(21,29)(22,31);(1,3)(2,8)(4,17,19,21,20,18,22,13)(5,7)(6,15)(9,26,28,23,27,32,31,29)(10,11)(12,30)(14,16)(24,25);(1,4)(2,9)(3,13)(5,21)(6,22)(7,20)(8,23)(10,29)(11,27)(12,31)(14,19)(15,17)(16,18)(24,28)(25,26)(30,32);(1,5,6,15,7,3,14,16)(2,10,12,30,11,8,24,25)(4,18,19,13,20,17,22,21)(9,26,28,23,27,32,31,29)Polycyclic presentation:
2^10|b^2=g,d^2=h,g^2=ij,h^2=j,b^a=bj,c^a=ce,c^b=cf,d^a=di,d^b=dghj,d^c=dhj,e^b=ej,e^d=ej,f^a=fj,f^b=fh,f^d=fh,g^c=ghj,g^f=gij,h^b=hi,h^c=hj,h^f=hj,i^c=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72306628, #72306731, #72306923, … |
| maximal normal subgroup of | #72306628, #72306731, #72306923, … |
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, 2, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);