#72341074

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5790 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=hj,d^2=fghi,e^2=i,b^a=bi,c^b=cgh,d^a=dhi,d^b=df,d^c=dgj,e^a=ei,e^b=egi,e^c=ehi,e^d=egi,f^a=fj,f^b=fi,f^c=fij,f^d=fij,f^e=fi,g^c=gj,g^d=gi,h^b=hj,h^d=hj

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341110, #72341169, …
maximal normal subgroup of	#72341083, #72341110, #72341169, …

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