#72284433

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
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2810 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)(3,7)(4,12)(5,6)(8,13)(9,14)(10,11)(15,16)(17,18)(19,24)(20,33)(21,36)(22,27)(23,28)(25,35)(26,37)(29,45)(30,42)(31,40)(32,41)(34,43)(38,47)(39,48)(44,46);(1,3,6,11)(2,7,5,10)(4,9,13,15)(8,14,12,16)(17,19,23,32)(18,24,28,41)(20,29,35,47)(21,30,37,48)(22,31,34,46)(25,38,33,45)(26,42,36,39)(27,44,43,40);(1,4)(2,8,5,12)(3,9)(6,13)(7,14,10,16)(11,15)(17,20)(18,25,28,33)(19,29)(21,22,37,34)(23,35)(24,38,41,45)(26,43)(27,36)(30,31,48,46)(32,47)(39,44)(40,42);(1,5,6,2)(3,10)(4,12,13,8)(7,11)(9,14)(15,16)(17,21)(18,26)(19,30)(20,34,35,22)(23,37)(24,39)(25,27,33,43)(28,36)(29,46,47,31)(32,48)(38,40,45,44)(41,42);(4,13)(7,10)(17,22,23,34)(18,27,28,43)(19,31,32,46)(20,21,35,37)(24,40,41,44)(25,36,33,26)(29,30,47,48)(38,42,45,39)

Polycyclic presentation:

2^10|b^2=i,c^2=f,d^2=ghij,e^2=j,b^a=bg,c^a=ch,d^a=dghi,d^b=dfgij,d^c=dhi,e^a=eh,e^b=efg,e^c=efghij,e^d=eh,f^a=fi,f^d=fi,f^e=fj,g^c=gij,g^d=gj,g^e=gj,h^b=hij,h^c=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285163
maximal normal subgroup of	#72285163

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