Finite Group Databasebeta

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#72341175

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#485685
commutator, Frattini, cosocle#379C24
prime radical#7266798C2
LCSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#379 ◁ G
LECSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1636 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1636 conjugacy classes of subgroups — too many to draw. Only its 121 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=gj,e^2=g,b^a=bgj,c^a=ci,c^b=ce,d^a=di,d^b=dhij,d^c=df,e^a=ej,e^b=eg,e^c=eg,e^d=eh,f^b=fh,f^c=fj,f^e=fij,g^d=gj,h^c=hi,h^e=hj,i^b=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341180
maximal normal subgroup of#72341180

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