Finite Group Databasebeta

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#72285166

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#142216
commutator, Frattini, cosocleC2×#379C23
LCSG ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ C25 ◁ C2×#379 ◁ G
LECSG ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

2541 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 2541 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=dhi,d^2=h,e^2=hij,b^a=bf,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dj,e^a=egi,e^b=ehi,e^c=eghi,e^d=eij,f^c=fj,g^a=gj,g^c=gi,g^d=gj,g^e=gj,h^b=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285384, #72285469
maximal normal subgroup of#72285384, #72285469

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