Finite Group Databasebeta

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#72284469

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#23900
commutator, Frattini, cosocleC2×#117C24
LCSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

3778 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 3778 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 40
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(11,19)(12,25)(13,28)(36,38);(1,2)(3,7)(4,6)(5,8)(9,10)(11,20,19,32)(12,26,25,16)(13,29,28,17)(14,18)(15,24)(21,38,33,36)(22,39)(23,40)(27,30)(31,35)(34,37);(1,2)(3,6)(4,7)(5,8)(9,10)(11,21)(12,16)(13,29)(14,18)(15,23)(17,28)(19,33)(20,36)(22,30)(24,40)(25,26)(27,39)(31,34)(32,38)(35,37);(2,6,8,7)(9,11,22,36,14,19,27,38)(10,15,30,37,18,31,39,40)(12,23,28,35,25,34,13,24)(16,21,17,20,26,33,29,32)

Polycyclic presentation:

2^10|b^2=g,d^2=efh,e^2=i,f^2=ij,b^a=bg,c^a=cg,c^b=cg,d^a=dh,d^b=df,d^c=de,e^a=eh,e^b=ehj,e^c=ei,f^a=fh,f^b=fhij,f^c=fj,g^d=gh,g^e=gj,g^f=gj,h^b=hj,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284565, #72285434, #72285435
maximal normal subgroup of#72284565, #72285434, #72285435

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