#72260685

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10745
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
prime radical	C₂×#2404	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

2613 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgj,d^2=j,g^2=j,b^a=bhij,c^a=ce,c^b=cd,d^a=dh,d^b=di,d^c=dij,e^b=eh,e^c=eij,e^d=ej,f^c=fij,g^a=gij,g^b=gi,g^c=gij,h^c=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260690, #72260773, #72260792
maximal normal subgroup of	#72260690, #72260773, #72260792

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