#72341104

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

2775 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)(3,10)(4,12,14,8)(5,7)(6,9)(11,15,13,16)(17,18,22,26)(19,23,29,39)(20,30)(21,33,36,25)(24,32)(27,37)(28,38,34,46)(31,44)(35,40)(41,48)(42,43)(45,47);(1,3)(2,7)(4,13,14,11)(5,6)(8,15,12,16)(9,10)(17,19,22,29)(18,23,26,39)(20,27)(21,34,36,28)(24,37)(25,38,33,46)(30,43)(31,45)(32,42)(35,41)(40,47)(44,48);(1,3)(2,7,9,10)(4,11,14,13)(5,6)(8,15)(12,16)(17,20,22,32)(18,24,26,30)(19,27)(21,35)(23,37)(25,40)(28,41,34,45)(29,42)(31,36)(33,44)(38,48,46,47)(39,43);(1,4,6,14)(2,8,9,12)(3,11)(5,13)(7,15)(10,16)(17,21)(18,25,26,33)(19,28,29,34)(20,31)(22,36)(23,38)(24,40,30,44)(27,41,42,45)(32,35)(37,47)(39,46)(43,48);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(19,29)(20,32)(21,36)(23,39)(24,30)(25,33)(27,42)(31,35)(37,43)(40,44)(41,45)(47,48)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=h,d^2=fij,e^2=ij,b^a=bgi,c^a=cf,c^b=cgh,d^a=dgh,d^b=dfhi,d^c=dfgj,e^b=egij,e^c=ehij,e^d=egij,f^a=fj,f^b=fi,f^c=fij,f^e=fij,g^c=gj,g^d=gij,h^a=hij,h^b=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341105, #72341110, #72341113
maximal normal subgroup of	#72341105, #72341110, #72341113

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