Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #131967 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#379 | C23 |
| prime radical | #6499869 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2849 ◁ G | |
| LECS | G ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
2798 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 2798 conjugacy classes of subgroups — too many to draw. Only its 89 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:
(1,2)(3,11)(4,15)(5,9)(6,10)(7,13)(8,17)(12,27)(14,28)(16,25)(18,22)(19,24)(20,26)(21,29)(23,31)(30,32);(1,3)(2,7,10,22)(4,12)(5,18)(6,14)(8,23)(9,11)(13,20)(15,21)(16,29)(17,30)(19,31)(24,27,25,32)(26,28);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,25)(10,15)(11,27)(13,29)(14,30)(16,20)(18,31)(22,23)(24,26)(28,32)Polycyclic presentation:
2^10|b^2=e,d^2=h,b^a=bd,c^a=chj,c^b=cf,d^a=dh,d^b=dg,d^c=dhj,e^a=eghj,e^c=ei,e^d=ei,f^a=fh,f^b=fi,g^b=gi,g^c=gj,g^d=gj,h^b=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72323228, #72324302, #72324303 |
| maximal normal subgroup of | #72323228, #72324302, #72324303 |
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, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);