Finite Group Databasebeta

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#72337628

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#32898
commutator, Frattini, cosocleC23×D4C24
prime radical#449492C2
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

4592 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 4592 conjugacy classes of subgroups — too many to draw. Only its 199 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) = 128

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=j,f^2=j,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^c=eh,e^d=ei,f^b=fhj,f^c=fj,f^d=fj,f^e=fj,g^b=gi,g^c=gj,h^a=hj

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72337630, #72337632, #72337668
maximal normal subgroup of#72337630, #72337632, #72337668

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