#72341109

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

5741 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)(3,10,12,7)(4,13,5,8)(6,9)(11,16)(14,15)(17,18)(19,23,29,38)(20,30,31,24)(21,25)(22,26)(27,39)(28,37,44,41)(32,36)(33,46,35,40)(34,42)(43,48)(45,47);(1,3,6,12)(2,7,9,10)(4,14,5,11)(8,16,13,15)(20,31)(21,34)(24,30)(27,32)(28,44)(33,35)(36,39)(43,45);(1,4)(2,8)(3,11)(5,6)(7,15)(9,13)(10,16)(12,14)(17,19)(18,23,26,38)(20,32,31,27)(21,28)(22,29)(24,39)(25,41,42,37)(30,36)(33,43,35,45)(34,44)(40,48)(46,47);(1,5,6,4)(2,8)(3,11)(7,15,10,16)(9,13)(12,14)(17,20,22,31)(18,24)(19,27)(21,35)(23,36,38,39)(25,40,42,46)(26,30)(28,45,44,43)(29,32)(33,34)(37,47)(41,48);(3,12)(4,5)(7,10)(8,13)(17,21,22,34)(18,25,26,42)(19,28,29,44)(20,33,31,35)(23,37,38,41)(24,40,30,46)(27,43,32,45)(36,47,39,48)

Polycyclic presentation:

2^10|a^2=g,b^2=j,c^2=gh,d^2=fhij,e^2=i,b^a=bf,c^a=cfh,c^b=cj,d^a=dfghi,d^b=dfij,d^c=dfgij,e^b=eghij,e^c=eghj,e^d=eghj,f^a=fj,f^c=fj,f^d=fj,f^e=fi,g^b=gj,g^c=gij,g^d=gij,h^a=hi,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341110, #72341145
maximal normal subgroup of	#72341110, #72341145

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