Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23984 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| prime radical | #347042 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
3273 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 3273 conjugacy classes of subgroups — too many to draw. Only its 137 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:
(1,2,6,10)(3,11,17,28)(4,18,13,26)(5,9,21,27)(7,22,24,19)(8,20,23,31)(12,30,14,15)(16,29,32,25);(1,2)(3,12,17,14)(4,8)(5,9)(6,10)(7,16,24,32)(11,15,28,30)(13,23)(18,31)(19,29,22,25)(20,26)(21,27);(1,3,13,30)(2,7,23,28)(4,19,6,16)(5,17,31,15)(8,25,10,14)(9,24,18,11)(12,26,29,27)(20,22,21,32);(1,2)(3,14)(4,8)(5,9)(6,10)(7,16)(11,22)(12,17)(13,23)(15,29)(18,31)(19,28)(20,26)(21,27)(24,32)(25,30)Polycyclic presentation:
2^10|a^2=g,b^2=ghj,c^2=eg,b^a=bg,c^a=ce,c^b=cf,d^a=dj,d^b=dghj,d^c=dj,e^a=ei,e^b=eh,e^c=ei,f^a=fhij,f^b=fij,f^c=fh,f^d=fij,f^e=fj,g^c=gi,h^c=hj58 conjugacy classes — view the full character table.
| maximal subgroup of | #72262851, #72262858, #72262875, … |
| maximal normal subgroup of | #72262851, #72262858, #72262875, … |
| maximal quotient of | #72338960, #72338969 |
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);;
gap> SetPower(coll, 2, F.7*F.8*F.10);;
gap> SetPower(coll, 3, F.5*F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
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.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);