#72340491

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	#245231	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2724 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)(3,9,11,6)(4,7)(5,8)(10,15,13,16)(12,14)(17,18)(19,23,29,38)(20,30,33,24)(21,25)(22,26)(27,36)(28,37,34,40)(31,44)(32,45,46,39)(35,41)(42,47)(43,48);(1,2)(3,6)(4,12,14,7)(5,8)(9,11)(10,15,13,16)(17,19)(18,23,26,38)(20,31,33,27)(21,34)(22,29)(24,36)(25,40,41,37)(28,35)(30,44)(32,42,46,43)(39,48)(45,47);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,15,12,16)(21,35)(28,34)(39,45)(47,48);(1,4,5,14)(2,7)(3,10,11,13)(6,15)(8,12)(9,16)(17,20,22,33)(18,24,26,30)(19,27,29,31)(21,32)(23,36,38,44)(25,39)(28,43)(34,42)(35,46)(37,48)(40,47)(41,45);(4,14)(7,12)(17,21)(18,25)(19,28)(20,32)(22,35)(23,37)(24,39)(26,41)(27,42)(29,34)(30,45)(31,43)(33,46)(36,47)(38,40)(44,48)

Polycyclic presentation:

2^10|a^2=h,b^2=fh,c^2=i,d^2=gij,b^a=bf,c^a=cg,d^a=dfi,d^b=dfgi,d^c=dgi,e^b=ej,e^c=efhj,e^d=ehij,f^a=fj,f^b=fj,f^c=fi,f^d=fi,g^a=gi,g^b=gi,g^e=gj,h^b=hj,h^c=hi,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340495
maximal normal subgroup of	#72340495

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