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#72225586

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#123275
commutator, Frattini, cosocleC8×C16C23
prime radical#492532C2
LCSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ G
LECSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

373 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,14)(3,16,17,52)(5,9,21,41)(6,28,22,10)(7,12,23,15)(8,31,36,18)(11,45,38,26)(13,48,44,25)(19,56,54,50)(20,51,30,53)(24,49,57,33)(27,39,58,42)(29,46,59,32)(34,60,37,55)(35,63,43,61)(40,64,47,62);(1,3,4,17)(2,8,14,36)(5,24,21,57)(6,29,22,59)(7,30,23,20)(9,16,41,52)(10,43,28,35)(11,46,38,32)(12,37,15,34)(13,49,44,33)(18,45,31,26)(19,48,54,25)(27,63,58,61)(39,53,42,51)(40,56,47,50)(55,62,60,64);(2,9,13,40,12,39,10,38,14,41,44,47,15,42,28,11)(3,17)(8,32,35,53,34,56,33,16,36,46,43,51,37,50,49,52)(18,31)(19,54)(20,30)(24,57)(29,59)(55,60)(61,63)

Polycyclic presentation:

2^10|a^2=j,b^2=j,c^2=dfj,d^2=fh,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=be,c^a=ce,c^b=cd,d^a=dg,d^b=df,e^a=eg,e^b=eg,f^a=fi,f^b=fh,g^a=gi,g^b=gi,h^a=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225587, #72225591
maximal normal subgroup of#72225587, #72225591

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