#72306986

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33517	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5029 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)(10,15)(12,24)(13,26)(16,31)(17,25)(18,35)(19,36)(20,38)(21,32)(22,39)(27,40)(28,33)(29,30)(34,37);(1,3)(2,5)(4,7)(6,8)(9,10,12,25,26,29,40,30,14,17,27,15,13,18,24,35)(11,19,21,37,39,33,38,28,23,34,20,36,22,16,32,31);(2,6)(3,7)(9,11)(10,16,17,33)(12,21)(13,22)(14,23)(15,28,25,31)(18,34,29,19)(20,27)(24,32)(26,39)(30,37,35,36)(38,40);(3,7)(5,8);(9,12,26,40,14,27,13,24)(10,15,29,35,17,25,18,30)(11,20,39,32,23,21,22,38)(16,31,19,37,33,28,34,36)

Polycyclic presentation:

2^10|b^2=fh,c^2=h,e^2=ij,f^2=ij,i^2=j,b^a=bf,c^a=cg,c^b=cgi,d^b=dgh,e^a=ei,e^b=ej,e^c=ej,f^a=fi,f^b=fj,g^b=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306996, #72307057, #72307438
maximal normal subgroup of	#72306996, #72307057, #72307438

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