Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #92866 |
| commutator | C8×C16 | C2×C8 |
| Frattini, cosocle | C2×#7052 | C22 |
| prime radical | C2×#12515 | C4 |
| LCS | G ▷ C8×C16 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#7052 ◁ G | |
| LECS | G ▷ C2×#7052 ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
507 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 512 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,11,5,13,19,30,7,15,20,29,6,27,18,31,9,17,22,28,26,32,24,12,8,16,21,10,25,14,4,23)(33,34,35,41,36,42,47,57,50,44,60,58,52,55,62,64,39,45,49,59,51,54,61,63,38,43,48,56,37,40,46,53);(2,10)(3,18)(4,22)(5,25)(6,26)(7,8)(11,16)(12,13)(14,23)(15,28)(17,29)(19,20)(21,24)(27,31)(30,32)(34,40)(35,46)(36,37)(38,50)(41,56)(42,43)(44,54)(45,55)(47,48)(49,62)(51,52)(57,63)(58,59)(60,61)Polycyclic presentation:
2^11|a^2=c,c^2=e,d^2=fghik,e^2=hj,f^2=h,g^2=ij,h^2=j,i^2=k,b^a=bd,c^b=cf,d^a=dg,d^b=dfg,d^c=djk,e^b=eh,f^a=fjk,f^b=fhj,g^a=gi,g^b=gijk,h^b=hj,i^a=ik,i^b=ik134 conjugacy classes — view the full character table.
| maximal subgroup of | #72227284 |
| maximal normal subgroup of | #72227284 |
| maximal quotient of | #72227272, #72227273 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.9*F.11);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
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.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);