#72239154

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#95
commutator	C₂⁴	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	C₂⁶	C₄²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5151 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efi,c^2=j,d^2=ij,e^2=j,f^2=i,b^a=bg,c^a=ch,d^a=dh,d^b=dhij,e^b=ei,f^a=fj,f^b=fi,g^a=gi,g^b=gj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72239177
Fitting subgroup of	#72239177
maximal subgroup of	#72239155, #72239177
maximal normal subgroup of	#72239155, #72239177

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