#72308153

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	#7623361	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20331 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3257 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=gj,d^2=f,e^2=hj,g^2=i,h^2=i,i^2=j,c^a=cg,c^b=cghij,d^c=di,e^a=ehj,e^b=eghi,e^c=ej,e^d=ei,f^c=fj,f^e=fj,g^a=gi,g^d=gj,h^a=hi,h^d=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308189, #72308468
maximal normal subgroup of	#72308189, #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, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
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);