#72306339

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#20609
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#7739265	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=ij,b^2=f,c^2=j,d^2=i,g^2=j,b^a=bg,c^b=ce,d^b=dgi,e^a=ei,e^b=eh,e^d=ei,f^a=fi,f^c=fh,f^d=fij,f^e=fj,g^b=gij,g^c=gij,g^e=gj,g^f=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306348, #72306355, #72306409, …
maximal normal subgroup of	#72306348, #72306355, #72306409, …

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