#72310879

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

2849 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=g,e^2=ij,c^a=cg,c^b=cf,d^b=dh,d^c=dgj,e^a=ehij,e^b=ei,e^c=efhj,e^d=efij,f^a=fi,f^c=fi,f^d=fj,g^b=gi,g^e=gj,h^c=hij,h^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310969, #72311205, #72311226, …
maximal normal subgroup of	#72310969, #72311205, #72311226, …

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