Finite Group Databasebeta

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#72283514

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152035
commutatorC2×C4×C8C22×C4
Frattini, cosocle#3147C23
prime radical#6357230C2
LCSG ▷ C2×C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ C4×#41 ◁ #60646 ◁ G
LECSG ▷ #3147 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1484 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 1484 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:

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

Polycyclic presentation:

2^10|b^2=dfj,c^2=ghj,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,e^b=eg,e^c=ej,f^a=fj,f^b=fj,f^c=fghj,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#72283689, #72284139
maximal normal subgroup of#72283689, #72284139

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.10);;
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, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
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*F.10);;
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);