#72310915

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(3,9)(4,11)(6,14)(10,13)(17,18)(19,24)(20,21)(22,37)(23,26)(25,40)(27,28)(29,41)(30,42)(31,34)(32,35)(33,36)(38,48)(39,47)(43,46)(44,45);(6,14)(7,12)(17,19)(18,24)(20,31)(21,34)(22,38)(23,30)(25,41)(26,42)(27,33)(28,36)(29,40)(32,43)(35,46)(37,48)(39,45)(44,47);(1,2)(3,6)(4,12,11,7)(5,8)(9,14)(10,16,13,15)(17,20)(18,21)(19,27,30,31)(22,39)(23,33)(24,34,42,28)(25,35)(26,36)(29,45,38,43)(32,40)(37,47)(41,46,48,44);(1,3)(2,6)(4,13,11,10)(5,9)(7,16,12,15)(8,14)(17,21)(18,20)(19,28,30,34)(22,35)(23,36)(24,31,42,27)(25,39)(26,33)(29,46,38,44)(32,37)(40,47)(41,45,48,43);(1,4,5,11)(2,7,8,12)(3,10,9,13)(6,15,14,16)(17,22,23,40)(18,25,26,37)(19,29,30,38)(20,32,33,39)(21,35,36,47)(24,41,42,48)(27,43,31,45)(28,44,34,46)

Polycyclic presentation:

2^10|c^2=g,d^2=g,e^2=ij,c^a=cg,c^b=cf,d^a=dgj,d^b=dh,e^a=efghij,e^b=ehi,e^c=ehi,e^d=efg,f^a=fi,f^c=fi,f^d=fj,f^e=fj,g^b=gi,g^e=gj,h^a=hi,h^c=hj,h^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310969, #72311321, #72340284, …
maximal normal subgroup of	#72310969, #72311321, #72340284, …

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