Finite Group Databasebeta

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#72339836

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152049
commutatorC4×M16C22×C4
Frattini, cosocle#3151C23
prime radical#6358059C2
LCSG ▷ C4×M16 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C4×#41 ◁ #60817 ◁ G
LECSG ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1151 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 1151 conjugacy classes of subgroups — too many to draw. Only its 81 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)(3,13)(4,19)(5,11)(6,10)(7,12)(8,16)(9,20)(14,29)(15,24)(17,26)(18,28)(21,22)(23,27)(25,31)(30,32);(1,3,2,8)(4,15,21,24)(5,17,10,28)(6,16,11,13)(7,18,12,26)(9,25,23,14)(19,29,22,31)(20,30,27,32);(3,14,16,15,18,31,17,30)(4,20)(8,24,28,25,26,32,13,29)(9,19)(21,27)(22,23)

Polycyclic presentation:

2^10|b^2=e,c^2=ghi,d^2=hi,f^2=j,g^2=ij,h^2=j,b^a=bf,c^b=cd,d^a=dij,d^b=dg,d^c=dj,e^c=egh,e^d=ehi,f^a=fj,f^b=fj,f^c=fi,f^d=fj,g^b=gh,g^e=gij,h^b=hi,h^e=hj,i^b=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339838, #72339877, #72339883
maximal normal subgroup of#72339838, #72339877, #72339883

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