Finite Group Databasebeta

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#72319717

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#147779
commutatorC2×#127C22×C4
Frattini, cosocleC22×#127C23
prime radical#7503372C2
LCSG ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G
LECSG ▷ C22×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

2638 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 2638 conjugacy classes of subgroups — too many to draw. Only its 147 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)(3,7)(4,15,17,8)(5,20)(6,9)(10,21)(11,25,29,31)(12,30)(13,23)(14,24)(16,22,28,32)(18,27,19,26);(1,3,6,14)(2,7,9,24)(4,16,19,29)(5,13)(8,25,27,32)(10,23)(11,17,28,18)(12,21)(15,31,26,22)(20,30);(1,4,6,17)(2,8,9,15)(3,11,14,29)(5,18,21,19)(7,22,24,32)(10,26,20,27)(12,28,13,16)(23,25,30,31)

Polycyclic presentation:

2^10|a^2=e,b^2=gj,c^2=j,h^2=j,b^a=beg,c^a=cd,c^b=cef,d^a=dj,d^b=dh,e^c=ej,f^b=fhj,f^c=fj,g^c=gh,g^d=gij,g^f=gij,h^a=hi,h^b=hi,h^c=hj,h^d=hj,h^f=hj,h^g=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72319727, #72319910
maximal normal subgroup of#72319727, #72319910

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