#72235229

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#119556
commutator	C₂³×D₄	C₂²×C₄
Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂³
prime radical	#35527	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

1544 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=dghj,c^2=dhi,d^2=j,g^2=j,b^a=bd,c^a=ce,c^b=cdef,d^a=dh,d^b=dhj,e^a=ehi,e^b=eh,e^c=ehj,e^d=ej,f^a=fhij,f^b=fhi,f^c=fj,g^b=ghj,g^c=ghi,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, #72235361
maximal normal subgroup of	#72235241, #72235361
maximal quotient of	#72235265, #72235372

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