Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139718 |
| commutator | C2×#117 | C22×C4 |
| Frattini, cosocle | C2×#535 | C23 |
| prime radical | #417774 | C2 |
| LCS | G ▷ C2×#117 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35222 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
2124 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 2124 conjugacy classes of subgroups — too many to draw. Only its 77 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 — 3 generators on 32 points:
(11,25)(12,17)(29,30)(31,32);(1,2)(3,11,13,29)(4,16,18,27)(5,20,21,9)(6,10)(7,12,22,31)(8,24)(14,25,15,30)(17,28,32,23)(19,26);(1,3,5,13,6,14,21,15)(2,7,20,22,10,23,9,28)(4,17,24,32,18,12,19,31)(8,25,27,29,26,11,16,30)Polycyclic presentation:
2^10|b^2=e,c^2=efghi,d^2=ij,h^2=j,b^a=bg,c^a=cef,c^b=cd,d^a=dg,d^b=dghj,d^c=dg,e^c=eghij,e^d=eij,f^b=fg,f^c=fghij,f^d=fi,g^c=gj,h^b=hi,h^c=hj,h^e=hj,h^f=hj,i^b=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72251733, #72251744, #72251754, … |
| maximal normal subgroup of | #72251733, #72251744, #72251754, … |
| maximal quotient of | #72338288, #72339641 |
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.5);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);