Finite Group Databasebeta

← Prev (#72287328) Next (#72287330) →

#72287329

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#130204
commutatorC2×M32C22×C4
Frattini, cosocleC2×#504C23
prime radical#7624863C2
LCSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33434 ◁ G
LECSG ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

837 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 837 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(2,9,15,23)(3,16)(4,21)(5,8)(7,31)(10,38)(11,34,17,45)(12,36,14,49)(13,50,52,61)(18,42)(20,56)(22,32,37,35)(24,33)(25,30)(26,44)(27,59)(29,41)(39,46,62,48)(40,54,64,47)(43,57)(51,53)(55,63,60,58);(1,2,3,11)(4,22,24,60)(5,14,28,46)(6,15,16,17)(7,32,20,58)(8,12,19,48)(9,33,34,59)(10,39,41,36)(13,51,47,31)(18,55,27,37)(21,23,42,45)(25,52,56,54)(26,50,38,40)(29,61,57,64)(30,35,53,63)(43,62,44,49);(1,2,10,40,30,52,18,45,5,12,42,55,31,32,38,39,6,15,43,64,7,13,24,34,8,14,33,60,25,35,57,62)(3,17,41,61,53,47,27,9,28,46,21,22,51,63,26,49,16,11,44,50,20,54,4,23,19,48,59,37,56,58,29,36)

Polycyclic presentation:

2^10|a^2=gj,b^2=d,c^2=defh,e^2=hij,f^2=j,h^2=ij,i^2=j,b^a=be,c^a=ce,c^b=cefj,d^a=dg,d^c=dj,e^a=eh,e^b=eghj,e^c=ej,e^d=ej,f^a=fg,f^b=fgj,f^d=fj,f^e=fj,g^b=gj,g^c=gj,h^a=hi,h^b=hi,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72287373, #72321336, #72342904
maximal normal subgroup of#72287373, #72321336, #72342904

GAP

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*F.10);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.8);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
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);