Finite Group Databasebeta

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#72283975

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152036
commutatorC2×C4×C8C22×C4
Frattini, cosocle#3147C23
prime radical#60647C22
LCSG ▷ C2×C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ C4×#41 ◁ #60647 ◁ G
LECSG ▷ #3147 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1307 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 1307 conjugacy classes of subgroups — too many to draw. Only its 87 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:

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

Polycyclic presentation:

2^10|b^2=dfj,c^2=ghi,d^2=j,e^2=hij,f^2=j,g^2=ij,h^2=j,b^a=bd,c^b=ce,d^a=dj,d^b=dj,d^c=dj,e^a=ej,e^b=eg,f^a=fj,f^b=fj,f^c=fgh,f^e=fhi,g^b=gh,g^f=gij,h^b=hi,h^f=hj,i^b=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284014, #72284107
maximal normal subgroup of#72284014, #72284107

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