#72313847

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	#6443328	C₂
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10)(11,15)(12,16)(13,25)(14,17)(18,26)(19,34)(20,37)(21,31)(22,30)(23,33)(24,35)(27,29)(28,32)(36,39)(38,40);(2,6)(3,4)(7,8)(11,19)(13,26)(15,29)(18,25)(20,36)(21,27)(22,38)(24,28)(30,39)(31,34)(32,35)(37,40);(2,7)(3,4)(6,8)(9,11)(10,15)(12,22)(13,27)(14,21)(16,30)(17,31)(18,34)(19,26)(20,23)(24,36)(25,29)(28,38)(32,40)(33,37)(35,39);(1,2,5,8)(3,6,4,7)(9,12,14,23)(10,16,17,33)(11,20)(13,28,26,24)(15,30)(18,35,25,32)(19,36)(21,22)(27,38)(29,39)(31,37)(34,40)

Polycyclic presentation:

2^10|d^2=hj,e^2=i,f^2=i,g^2=j,b^a=bhi,c^b=cg,d^a=dhi,d^b=df,d^c=de,e^a=ej,e^b=eh,e^c=ei,e^d=eij,f^b=fi,f^c=fi,f^d=fi,g^a=gj,g^b=gj,g^c=gj,g^d=ghij,g^e=gj,h^c=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313853, #72313956, #72313966
maximal normal subgroup of	#72313853, #72313956, #72313966

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