#72341146

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

5790 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:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=i,d^2=h,e^2=ij,b^a=bf,d^a=dh,d^b=dfij,d^c=dg,e^a=ehj,e^b=eg,e^c=egij,e^d=ehi,f^d=fi,f^e=fij,g^a=gj,g^d=gj,h^b=hi,h^c=hj

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341176
maximal normal subgroup of	#72341176

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