#72285994

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

2812 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)(3,9,11,6)(4,12)(5,8)(7,14)(10,15,13,16)(17,18)(19,27,30,23)(20,24,32,41)(21,33)(22,26)(25,36)(28,38)(29,39,34,43)(31,42,35,40)(37,45)(44,47)(46,48);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,39)(26,27)(29,36)(31,44)(32,45)(33,43)(35,46)(38,41)(40,48)(42,47);(1,4)(2,7,8,12)(3,10)(5,14)(6,15,9,16)(11,13)(17,20,22,32)(18,24)(19,28,30,45)(21,35)(23,37)(25,42,33,40)(26,41)(27,38)(29,44)(31,36)(34,46)(39,47,43,48);(3,11)(7,12)(10,13)(15,16)(17,20)(18,24)(19,28)(21,35)(22,32)(23,38)(25,42)(26,41)(27,37)(29,46)(30,45)(31,36)(33,40)(34,44)(39,47)(43,48);(3,11)(4,14)(6,9)(7,12)(10,13)(15,16)(17,21)(18,25)(19,29)(20,31)(22,36)(23,39)(24,40)(26,33)(27,43)(28,44)(30,34)(32,35)(37,47)(38,48)(41,42)(45,46)

Polycyclic presentation:

2^10|a^2=f,b^2=ij,c^2=fgj,b^a=bfg,c^a=cfh,c^b=cij,d^a=dhj,d^b=dfgij,d^c=dfgj,e^a=ej,e^b=efhi,e^c=efij,e^d=ej,f^b=fi,f^c=fj,f^d=fj,g^a=gi,g^b=gi,g^c=gj,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286044, #72341133
maximal normal subgroup of	#72286044, #72341133

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