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#72284593

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#485605
commutator, Frattini, cosocle#379C24
prime radical#151130C2
LCSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2280 ◁ G
LECSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1744 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1744 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=f,c^2=j,d^2=j,f^2=h,b^a=bf,c^b=ch,d^a=de,d^b=dei,d^c=dij,e^b=eg,f^a=fh,f^d=fgj,f^e=fij,g^a=gi,g^b=gij,g^f=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284770
maximal normal subgroup of#72284770

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