#72318630

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4754 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,23)(13,18)(14,19)(15,22)(16,25)(20,32)(21,36)(24,30)(26,37)(27,34)(28,31)(29,38)(33,40)(35,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,24)(13,21)(14,22)(16,32)(17,19)(18,31)(20,25)(23,30)(26,39)(27,40)(28,29)(33,34)(35,37)(36,38);(1,3)(2,5,6,7)(4,8)(9,12,13,27,14,26,29,25)(10,16,18,37,19,34,38,23)(11,20,28,39,22,40,21,24)(15,30,36,33,17,35,31,32);(9,11)(10,17)(12,24)(13,28)(14,22)(15,19)(16,33)(18,31)(20,27)(21,29)(23,30)(25,40)(26,39)(32,34)(35,37)(36,38)

Polycyclic presentation:

2^10|c^2=efj,g^2=j,b^a=bhj,c^a=ce,c^b=cef,d^c=dg,e^b=eh,e^c=eh,e^d=ei,f^a=fh,f^b=fhj,f^c=fh,f^d=fi,f^e=fj,g^a=gij,g^c=gj,g^d=gj,g^e=gj,g^f=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318703, #72318854, #72319038, …
maximal normal subgroup of	#72318703, #72318854, #72319038, …

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