#72341167

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

5947 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)(4,8)(5,9)(7,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,33)(27,42)(28,44)(29,39)(31,37)(32,40)(34,38)(35,45)(36,41)(43,48)(46,47);(1,3)(2,6)(4,12,13,10)(5,11)(7,9)(8,15,16,14)(17,19)(18,23)(20,31,33,27)(21,34)(22,29)(24,37,30,42)(25,38)(26,39)(28,36)(32,46,35,43)(40,48,45,47)(41,44);(1,3)(2,6)(4,12)(5,11)(7,9)(8,15)(10,13)(14,16)(24,30)(27,31)(40,45)(43,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,13)(14,16)(17,20,22,33)(18,24)(19,27)(21,35,36,32)(23,37,39,42)(25,40)(26,30)(28,43)(29,31)(34,46)(38,48,44,47)(41,45);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,21,22,36)(18,25)(19,28)(20,32,33,35)(23,38,39,44)(24,40)(26,41)(27,43)(29,34)(30,45)(31,46)(37,47,42,48)

Polycyclic presentation:

2^10|b^2=g,d^2=fg,e^2=fg,b^a=bg,c^a=cg,c^b=cg,d^a=dgi,d^b=df,d^c=dfgj,e^a=egh,e^b=efh,e^c=ei,e^d=efg,f^a=fj,f^b=fj,f^d=fj,f^e=fi,g^d=gj,g^e=gi,h^b=hij,h^c=hi,h^d=hi,h^e=hij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341188
maximal normal subgroup of	#72341188

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