#72307681

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#8023
commutator	C₂²×C₈	C₂×C₄²
Frattini, cosocle	C₂³×M₁₆	C₂³
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂³×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=ef,c^2=i,d^2=j,e^2=i,i^2=j,b^a=bdeij,c^a=cij,d^b=dg,e^a=egij,e^b=egh,e^d=ej,f^a=fgh,f^b=fgh,g^a=gj,g^b=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307689, #72307805, #72308149
maximal normal subgroup of	#72307689, #72307805, #72308149

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