#72306964

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#391
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	C₂×#1985
prime radical	C₂⁵	C₂×M₁₆
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2281 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=f,e^2=j,f^2=i,i^2=j,b^a=bg,c^a=ce,c^b=ci,d^c=dgh,e^b=ej,f^a=fj,f^b=fj,g^c=gj,h^c=hj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306996, #72307077, #72307284
maximal normal subgroup of	#72306996, #72307077, #72307284

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