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 | #139504 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #2308 | C23 |
| prime radical | #417824 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #35346 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
2176 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 2176 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:
(8,22)(11,24)(13,29)(20,31)(23,25)(27,30);(1,2)(3,11)(4,8)(5,17)(6,10)(7,13)(9,19)(12,26)(14,24)(15,25)(16,22)(18,23)(20,27)(21,29)(28,32)(30,31);(1,3,4,12)(2,7,8,20)(5,13,18,30)(6,14,16,28)(9,24,23,32)(10,21,22,31)(11,25,26,17)(15,27,19,29)Polycyclic presentation:
2^10|c^2=d,e^2=hj,b^a=bf,c^a=cgi,c^b=ce,d^a=di,d^b=dg,e^a=eh,e^b=ehj,e^c=egh,e^d=eij,f^c=fh,f^d=fj,g^c=gi,g^e=gj,h^c=hj,i^b=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72237753, #72237757, #72237770, … |
| maximal normal subgroup of | #72237753, #72237757, #72237770, … |
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, 3, F.4);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);