Finite Group Databasebeta

← Prev (#72235220) Next (#72235222) →

#72235221

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#119556
commutatorC22×#41C22×C4
Frattini, cosocleC22×H(5,𝔽2)C23
prime radical#35527C22
LCSG ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1

1564 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 119 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=dei,c^2=degh,g^2=j,b^a=be,c^a=cg,c^b=cdf,d^b=dh,d^c=di,e^a=eh,e^b=ehj,e^c=ehj,f^a=fj,f^b=fi,f^c=fhij,f^e=fj,g^a=gij,g^b=gh,g^c=ghi,g^d=gj,g^e=gj,g^f=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235241, #72338959, #72338965, …
maximal normal subgroup of#72235241, #72338959, #72338965, …
maximal quotient of#72235265

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