Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #133121 |
| commutator | C2×#117 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | #6435072 | C2 |
| LCS | G ▷ C2×#117 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 1926 conjugacy classes of subgroups — too many to draw. Only its 87 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)(3,14,18,44)(4,10)(5,11)(6,26)(7,12)(8,13)(9,17,32,29)(15,42,47,64)(16,43,23,56)(19,50,53,60)(20,34)(21,36)(22,37)(24,40)(25,41)(27,35)(28,39)(30,46,59,57)(31,49,38,48)(33,52,61,51)(45,63)(54,58)(55,62);(1,3,6,17)(2,9,12,14,13,32,26,44)(4,19,21,51)(5,23,24,49)(7,29,8,18)(10,33,35,60,37,61,36,50)(11,38,39,43,41,31,40,56)(15,45,46,20)(16,28,48,25)(22,53,27,52)(30,58,42,62,59,63,64,34)(47,54,57,55);(1,4)(2,10)(3,15,18,47)(5,20)(6,27)(7,21)(8,22)(9,30,32,59)(11,34)(12,36)(13,37)(14,42,44,64)(16,19,23,53)(17,46,29,57)(24,54)(25,55)(26,35)(28,45)(31,33,38,61)(39,63)(40,58)(41,62)(43,50,56,60)(48,51,49,52)Polycyclic presentation:
2^10|a^2=h,b^2=e,c^2=h,d^2=j,e^2=hi,f^2=hj,b^a=bf,c^b=cd,d^b=dh,e^a=eg,e^c=ehj,e^d=ej,f^a=fh,f^b=fgh,f^e=fij,g^b=gi,g^e=gj,g^f=gj,h^b=hj,i^a=ij,i^b=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72285276, #72285412, #72285936, … |
| maximal normal subgroup of | #72285276, #72285412, #72285936, … |
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.8);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);