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 | #133121 |
| commutator | C2×#117 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | #6435070 | C2 |
| LCS | G ▷ C2×#117 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35272 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
2092 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 2092 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(2,7,10,20)(8,18,21,30)(9,19,23,32)(11,24,22,31);(1,2)(3,11)(4,9)(5,7,17,20)(6,10)(8,12,21,25)(13,24,27,31)(14,22)(15,19,28,32)(16,23)(18,29)(26,30);(1,3)(2,8,10,21)(4,12)(5,13)(6,14)(7,18,20,30)(9,22,23,11)(15,26)(16,25)(17,27)(19,31,32,24)(28,29)Polycyclic presentation:
2^10|a^2=gh,b^2=e,c^2=gh,d^2=ghj,f^2=j,b^a=bf,c^b=cd,d^b=dg,d^c=dgh,e^a=egh,e^c=ehj,e^d=ei,f^b=fghj,f^e=fj,g^b=gi,g^d=gj,h^b=hij,h^d=hj,i^c=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72339016, #72339018, #72339045 |
| maximal normal subgroup of | #72339016, #72339018, #72339045 |
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.7*F.8);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);