Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2840 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | C22×H(3,𝔽4) | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3022 conjugacy classes of subgroups; only the 491 normal subgroups are shown.
This group has 3022 conjugacy classes of subgroups — too many to draw. Only its 491 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 — 5 generators on 64 points:
(1,2)(3,8)(4,19,23,9)(5,25,28,10)(6,11)(7,13)(12,30)(14,43,20,35)(15,47,26,39)(16,42)(17,33)(18,34)(21,52,55,36)(22,38,24,37)(27,41,29,40)(31,46,60,45)(32,49,61,48)(44,62,53,63)(50,56,57,54)(51,64,58,59);(1,3)(2,8)(4,20,23,14)(5,26,28,15)(6,18)(7,17)(9,35,19,43)(10,39,25,47)(11,34)(12,42)(13,33)(16,30)(21,53,55,44)(22,46,24,45)(27,49,29,48)(31,37,60,38)(32,40,61,41)(36,63,52,62)(50,59,57,64)(51,56,58,54);(1,4)(2,9)(3,14)(5,21)(6,22)(7,24)(8,31)(10,36)(11,37)(12,19)(13,38)(15,44)(16,20)(17,46)(18,45)(23,30)(25,57)(26,58)(27,56)(28,54)(29,55)(32,59)(33,35)(34,43)(39,64)(40,50)(41,52)(42,60)(47,63)(48,51)(49,53)(61,62);(1,5)(2,10)(3,15)(4,21)(6,29)(7,28)(8,32)(9,36)(11,41)(12,40)(13,25)(14,44)(16,48)(17,26)(18,49)(19,50)(20,51)(22,55)(23,56)(24,54)(27,30)(31,59)(33,61)(34,39)(35,62)(37,52)(38,57)(42,47)(43,64)(45,53)(46,58)(60,63);(4,22)(5,27)(9,37)(10,40)(14,45)(15,48)(19,38)(20,46)(21,54)(23,24)(25,41)(26,49)(28,29)(31,43)(32,47)(35,60)(36,57)(39,61)(44,58)(50,52)(51,53)(55,56)(59,62)(63,64)Polycyclic presentation:
2^10|a^2=h,b^2=h,b^a=bh,c^a=cg,c^b=cf,d^a=dfg,d^b=dg,e^c=ei,e^d=eij,f^a=fi,f^b=fij,g^a=gij,g^b=gj,h^c=hij,h^d=hj55 conjugacy classes — view the full character table.
| Fitting subgroup of | #72340953 |
| maximal subgroup of | #72228243, #72249481, #72340953, … |
| maximal normal subgroup of | #72228243, #72249481, #72340953, … |
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.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);