#72341080

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=ij,b^2=gij,c^2=fhj,d^2=fhi,e^2=i,b^a=bg,c^a=cf,c^b=ch,d^a=dgj,d^b=dfghi,d^c=dfhij,e^a=egj,e^b=egj,e^c=egh,e^d=ehi,f^b=fi,f^c=fij,f^d=fi,f^e=fi,g^c=gj,g^d=gij,h^a=hij,h^b=hj,h^c=hij,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341103, #72341124
maximal normal subgroup of	#72341083, #72341103, #72341124

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