#72263155

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21746
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#4420586	C₂
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=ef,d^2=i,b^a=be,c^a=cghi,c^b=cghi,d^a=dhj,d^b=dghij,e^a=ei,e^b=eh,e^c=eij,e^d=ej,f^a=fhi,f^b=fh,f^d=fi,f^e=fj,g^b=gi,h^b=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263157, #72263166, #72263232, …
maximal normal subgroup of	#72263157, #72263166, #72263232, …

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