Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10725 |
| commutator | C23×C4 | C2×C42 |
| Frattini, cosocle | C24×D4 | C23 |
| prime radical | C23×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×#127 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1 | |
5177 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 5177 conjugacy classes of subgroups — too many to draw. Only its 217 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,31)(30,32);(1,3)(2,6)(4,12,13,10)(5,11)(7,15,16,14)(8,9)(17,19)(18,22)(20,26,29,32)(21,27)(23,25,28,31)(24,30);(1,4)(2,7)(3,10)(5,13)(6,14)(8,16)(9,15)(11,12)(20,29)(22,30)(23,28)(26,32)Polycyclic presentation:
2^10|a^2=g,b^2=e,d^2=j,b^a=bd,c^b=cf,d^a=di,d^b=dh,d^c=dj,e^a=ehj,e^c=eij,e^d=ej,f^a=fj,f^b=fij,g^b=gij,h^b=hj76 conjugacy classes — view the full character table.
| maximal subgroup of | #72266170, #72266237, #72266721 |
| maximal normal subgroup of | #72266170, #72266237, #72266721 |
| maximal quotient of | #72310558 |
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.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);