#72341081

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

5921 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=gi,c^2=fgh,d^2=fij,e^2=ij,b^a=bg,c^b=cf,d^a=dfhi,d^b=dh,d^c=dgh,e^a=egj,e^b=egj,e^c=eghij,e^d=egh,f^a=fi,f^b=fi,f^c=fij,f^e=fij,g^c=gi,g^d=gj,h^a=hi,h^b=hj,h^c=hj,h^d=hi

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341091, #72341103, …
maximal normal subgroup of	#72341083, #72341091, #72341103, …

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