#72284591

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂²×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3721 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)(3,9,11,4)(5,13,15,7)(6,8)(10,16)(12,14)(17,18,22,26)(19,23,29,36)(20,24)(21,32)(25,33)(27,34)(28,39)(30,40,45,37)(31,38)(35,43)(41,48,44,46)(42,47);(1,3)(2,4)(5,14)(6,11)(7,16)(8,9)(10,15)(12,13)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,39)(26,36)(30,44)(31,42)(32,35)(33,43)(37,46)(38,47)(40,48)(41,45);(1,4)(2,3,8,11)(5,12,15,16)(6,9)(7,10)(13,14)(17,20,22,31)(18,24)(19,27,29,42)(21,30,33,45)(23,34)(25,40)(26,38)(28,41,43,44)(32,37)(35,48)(36,47)(39,46);(1,5,6,15)(2,7)(3,10)(4,12,9,16)(8,13)(11,14)(17,21,22,33)(18,25,26,32)(19,28,29,43)(20,30)(23,35,36,39)(24,37)(27,41)(31,45)(34,46)(38,40)(42,44)(47,48);(2,8)(4,9)(5,15)(10,14)(18,26)(19,29)(24,38)(25,32)(27,42)(28,43)(37,40)(41,44)

Polycyclic presentation:

2^10|a^2=f,c^2=fg,d^2=fhj,b^a=bh,c^a=cg,c^b=ch,d^a=dfgi,d^b=dgh,d^c=dfghj,e^a=ei,e^b=eij,e^c=ej,e^d=ej,f^b=fj,f^c=fi,f^d=fi,g^a=gi,g^c=gi,g^d=gij,h^a=hj,h^c=hj,h^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72285517, #72340246, …
maximal normal subgroup of	#72284709, #72285517, #72340246, …

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