#72314107

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	#7622812	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4207 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=h,e^2=fgj,h^2=i,i^2=j,c^b=cg,d^a=dgj,d^b=dg,d^c=dh,e^a=egj,e^d=egi,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, #72314371, #72314497
maximal normal subgroup of	#72314160, #72314371, #72314497

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