Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10336 |
| commutator | C23×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C23×#41 | C23 |
| socle | C2 | #121408 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
1999 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 1999 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,11,7,12,13,24)(3,8,16,27,17,28,35,50)(4,9,20,31,21,32,36,51)(5,10,22,33,23,34,37,52)(14,25,39,54,40,55,61,63)(15,41,42,57,43,26,44,56)(18,29,45,58,46,59,62,64)(19,47,48,53,49,30,38,60);(1,3,13,35,7,17,6,16)(2,8,24,50,12,28,11,27)(4,18,20,45,21,46,36,62)(5,15,37,44,23,43,22,42)(9,29,31,58,32,59,51,64)(10,26,52,57,34,41,33,56)(14,38,39,19,40,48,61,49)(25,53,54,30,55,60,63,47);(1,4,6,20,7,21,13,36)(2,9,11,31,12,32,24,51)(3,14,35,61,17,40,16,39)(5,19,22,48,23,49,37,38)(8,25,50,63,28,55,27,54)(10,30,33,60,34,47,52,53)(15,18,44,62,43,46,42,45)(26,29,57,64,41,59,56,58)Polycyclic presentation:
2^10|a^2=e,b^2=eij,c^2=ehi,d^2=j,e^2=j,b^a=bf,c^a=cg,c^b=cd,d^a=dh,f^c=fhi,f^d=fj,g^b=gi,g^d=gj,h^b=hj,h^c=hj,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72317504, #72317598, #72317737 |
| maximal normal subgroup of | #72317504, #72317598, #72317737 |
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.5);;
gap> SetPower(coll, 2, F.5*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.8*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, 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.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);