#72308302

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	#7623308	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=h,d^2=f,g^2=j,h^2=i,i^2=j,c^a=ch,c^b=cg,d^c=dij,e^a=egi,e^c=egij,e^d=ej,f^c=fj,g^a=gj,h^a=hi,h^b=hj,h^d=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308337, #72308468
maximal normal subgroup of	#72308337, #72308468

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