Finite Group Databasebeta

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#72285683

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#238077
commutator, Frattini, cosocleC23×D4C24
prime radical#35346C22
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

1974 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1974 conjugacy classes of subgroups — too many to draw. Only its 175 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=ghij,d^2=g,b^a=be,c^a=cf,c^b=cgh,d^a=dfhj,d^b=dg,d^c=dhi,e^b=ei,e^c=eh,e^d=ehj,f^b=fhij,f^c=fi,f^d=fij,g^a=gi,g^c=gj,g^e=gj,h^b=hj,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285690, #72285757, #72285834, …
maximal normal subgroup of#72285690, #72285757, #72285834, …

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