Finite Group Databasebeta

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#72338733

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#485385
commutator, Frattini, cosocle#379C24
prime radical#485611C2
LCSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G
LECSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1490 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1490 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=gj,c^2=h,e^2=ghi,b^a=be,c^a=cefghj,c^b=chi,d^a=dghij,d^c=dj,e^a=eg,e^b=egh,e^c=egi,f^b=fhi,f^c=fhj,f^d=fj,f^e=fi,g^a=gi,g^e=gj,h^a=hij,h^b=hj,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338745
maximal normal subgroup of#72338745

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