Finite Group Databasebeta

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#72307487

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#18178
commutator, Frattini, cosocleC2×#97C24
LCSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

2186 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 2186 conjugacy classes of subgroups — too many to draw. Only its 147 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:

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

Polycyclic presentation:

2^10|a^2=gh,b^2=f,c^2=gh,e^2=h,f^2=i,i^2=j,b^a=bf,c^b=cij,d^a=dgh,d^b=de,d^c=dgh,e^a=eg,e^b=egh,e^c=ej,e^d=eh,f^a=fi,f^c=fj,f^d=fg,f^e=fj,g^b=gj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72307513, #72307671, #72307751
maximal normal subgroup of#72307513, #72307671, #72307751

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