#72314121

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623307	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=f,d^2=h,e^2=fgi,h^2=i,i^2=j,b^a=bg,c^a=chij,c^b=cg,d^b=dg,d^c=dh,e^a=eij,e^c=ei,e^d=egi,f^a=fj,f^d=fj,h^c=hi,h^e=hj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314160, #72314408, #72314521
maximal normal subgroup of	#72314160, #72314408, #72314521

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