#72285160

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

2827 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,15)(13,16)(21,34)(24,31)(25,35)(29,43)(32,45)(36,40)(37,39)(41,46);(1,2)(3,9,11,7)(4,12,14,8)(5,6)(10,16)(13,15)(17,18)(19,27)(20,31,33,24)(21,35)(22,26)(23,30)(25,34)(28,40,42,36)(29,37)(32,44,45,38)(39,43)(41,48,46,47);(1,3)(2,7)(4,13,14,10)(5,9)(6,11)(8,15,12,16)(17,19)(18,23,26,27)(20,28,33,42)(21,29)(22,30)(24,36)(25,39,35,37)(31,40)(32,46,45,41)(34,43)(38,47)(44,48);(1,4,6,14)(2,8,5,12)(3,10)(7,15)(9,16)(11,13)(17,20,22,33)(18,24)(19,28,30,42)(21,32,34,45)(23,36)(25,38)(26,31)(27,40)(29,41,43,46)(35,44)(37,47)(39,48);(1,5,6,2)(3,9,11,7)(4,8,14,12)(10,15,13,16)(17,21,22,34)(18,25,26,35)(19,29,30,43)(20,32,33,45)(23,37,27,39)(24,38,31,44)(28,41,42,46)(36,47,40,48)

Polycyclic presentation:

2^10|b^2=h,c^2=f,d^2=fhij,e^2=ij,b^a=bh,c^b=cg,d^a=dfhj,d^b=dfghi,d^c=dghij,e^a=eij,e^b=ehij,e^c=ef,e^d=ej,f^b=fi,f^d=fij,g^a=gj,g^b=gj,g^c=gi,g^d=gi,g^e=gij,h^c=hj,h^d=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185, #72341083, #72341091, …
maximal normal subgroup of	#72286185, #72341083, #72341091, …

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