#72340487

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	#251860	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2736 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)(7,14)(9,11)(10,16)(13,15)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,37)(27,39)(28,40)(29,38)(31,45)(32,43)(35,42)(36,41)(44,48)(46,47);(1,2)(3,9,11,6)(4,7,14,12)(5,8)(10,16)(13,15)(17,19)(18,23)(20,31)(21,35,37,28)(22,29)(24,39)(25,42,34,40)(26,38)(27,33)(30,45)(32,46,36,44)(41,47,43,48);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,15,12,16)(23,38)(24,30)(25,34)(47,48);(1,4)(2,7,8,12)(3,10,11,13)(5,14)(6,15)(9,16)(17,20)(18,24,26,30)(19,27)(21,36,37,32)(22,33)(23,39,38,45)(25,43)(28,44,35,46)(29,31)(34,41)(40,47)(42,48);(4,14)(6,9)(7,12)(15,16)(17,21,22,37)(18,25,26,34)(19,28,29,35)(20,32,33,36)(23,40,38,42)(24,41,30,43)(27,44,31,46)(39,47,45,48)

Polycyclic presentation:

2^10|b^2=f,c^2=i,d^2=g,e^2=j,b^a=bf,c^a=cfh,c^b=cfg,d^a=dfh,d^b=dfgj,d^c=dfg,e^a=ehj,e^b=eh,e^c=efg,e^d=ehi,f^c=fi,f^d=fi,f^e=fj,g^a=gij,g^b=gi,g^c=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, 2, F.6);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, 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);