#72340834

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#147561
commutator	C₂²×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#532	C₂³
prime radical	C₂×#3498	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=e,c^2=dfij,e^2=j,g^2=i,h^2=j,b^a=bd,c^a=cdf,c^b=cdfg,d^b=dij,d^c=dj,e^a=eij,e^c=eh,e^d=ej,f^c=fj,g^a=gi,g^b=gh,g^e=gij,h^a=hj,h^b=hi,h^e=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340839, #72340844, #72340859
maximal normal subgroup of	#72340839, #72340844, #72340859

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