#72341079

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

5839 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)(3,6)(4,12)(5,8)(7,13)(9,15)(10,14)(11,16)(17,18)(19,24)(20,25)(21,26)(22,35)(23,27)(28,37)(29,42)(30,40)(31,39)(32,41)(33,38)(34,44)(36,43)(45,48)(46,47);(1,3)(2,6)(4,10,13,9)(5,11)(7,14,12,15)(8,16)(17,19)(18,24)(20,30)(21,33,34,29)(22,28)(23,31)(25,40)(26,42,44,38)(27,39)(32,46,36,45)(35,37)(41,48,43,47);(1,4)(2,7)(3,9)(5,13)(6,14)(8,12)(10,11)(15,16)(17,20)(18,25)(19,28,31,30)(21,32,34,36)(22,23)(24,37,39,40)(26,41,44,43)(27,35)(29,46)(33,45)(38,48)(42,47);(1,4)(2,7,8,12)(3,10,11,9)(5,13)(6,15)(14,16)(17,21)(18,26)(19,29)(20,32,22,36)(23,34)(24,38)(25,41,35,43)(27,44)(28,45,30,46)(31,33)(37,47,40,48)(39,42);(4,13)(7,12)(9,10)(14,15)(17,22,23,20)(18,25,27,35)(19,30,31,28)(21,32,34,36)(24,37,39,40)(26,43,44,41)(29,46,33,45)(38,47,42,48)

Polycyclic presentation:

2^10|b^2=g,c^2=h,d^2=f,e^2=i,b^a=bg,c^a=cj,c^b=cf,d^a=dgh,d^b=dfgh,d^c=dfh,e^a=ei,e^b=ei,e^c=ehij,e^d=ehj,f^a=fj,f^b=fj,f^c=fi,f^e=fi,g^c=gj,g^d=gij,h^b=hi,h^d=hi

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341169, #72341188
maximal normal subgroup of	#72341083, #72341169, #72341188

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