#72341100

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

5871 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)(3,6)(4,12,14,7)(5,9)(8,11)(10,15,13,16)(17,18)(19,24,32,40)(20,25,35,42)(21,36,34,26)(22,27)(23,28)(29,37)(30,45)(31,48,46,39)(33,43)(38,47)(41,44);(1,3,5,11)(2,6,9,8)(4,13,14,10)(7,16,12,15)(17,19,23,32)(18,24)(20,29,35,44)(21,30)(22,31)(25,41)(26,38,36,45)(27,39,43,48)(28,40)(33,46)(34,47)(37,42);(1,4,5,14)(2,7,9,12)(3,10,11,13)(6,15,8,16)(17,20,23,35)(18,25)(19,29,32,44)(21,22)(24,37)(26,27,36,43)(28,42)(30,46)(31,47)(33,34)(38,39,45,48)(40,41);(1,3)(2,8)(4,13,14,10)(5,11)(6,9)(7,16,12,15)(17,21,23,34)(18,26)(19,30)(20,33)(22,35)(24,38,40,45)(25,27,42,43)(28,36)(29,31,44,46)(32,47)(37,39)(41,48);(2,9)(3,11)(17,22,23,33)(18,27,28,43)(19,31,32,46)(20,34,35,21)(24,39,40,48)(25,36,42,26)(29,30,44,47)(37,45,41,38)

Polycyclic presentation:

2^10|a^2=g,b^2=fgi,c^2=fgi,d^2=fj,e^2=j,b^a=bf,c^a=ch,c^b=cfgij,d^a=dfgi,d^b=dgij,d^c=dhj,e^a=egij,e^b=egi,e^c=eh,e^d=egij,f^a=fj,f^b=fj,f^c=fi,f^e=fj,g^b=gj,g^c=gi,g^d=gj,h^a=hi,h^b=hij,h^c=hj,h^d=hi

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341103, #72341110, #72341121
maximal normal subgroup of	#72341103, #72341110, #72341121

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