#72285402

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

2784 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)(3,10,13,7)(4,5)(6,9)(8,15)(11,12,14,16)(17,18)(19,24)(20,33,36,25)(21,26)(22,27,34,41)(23,28)(29,45,48,37)(30,38)(31,39,46,44)(32,40)(35,43)(42,47);(1,3,6,13)(2,7,9,10)(4,14,15,11)(5,12,8,16)(17,19,23,32)(18,24,28,40)(20,29,36,48)(21,30,35,47)(22,31,34,46)(25,37,33,45)(26,42,43,38)(27,44,41,39);(1,4,6,15)(2,8)(3,11,13,14)(5,9)(7,12)(10,16)(17,20,23,36)(18,25,28,33)(19,29,32,48)(21,22)(24,37,40,45)(26,41)(27,43)(30,31)(34,35)(38,44)(39,42)(46,47);(1,5,6,8)(2,4,9,15)(3,12)(7,14)(10,11)(13,16)(17,21,23,35)(18,26,28,43)(19,30,32,47)(20,34)(22,36)(24,38,40,42)(25,41)(27,33)(29,46)(31,48)(37,44)(39,45);(2,9)(3,13)(4,15)(5,8)(11,14)(12,16)(17,22,23,34)(18,27,28,41)(19,31,32,46)(20,35,36,21)(24,39,40,44)(25,26,33,43)(29,47,48,30)(37,38,45,42)

Polycyclic presentation:

2^10|a^2=g,b^2=ij,c^2=fij,d^2=gij,e^2=j,b^a=bf,c^a=cfgh,c^b=ci,d^a=dg,d^b=dfghi,d^c=dhij,e^a=efghij,e^b=efghi,e^c=efgi,e^d=ehi,f^a=fi,f^d=fij,f^e=fj,g^b=gi,g^c=gj,g^e=gj,h^a=hij,h^b=hi,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526, #72341097, #72341103, …
maximal normal subgroup of	#72285526, #72341097, #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.7);;
gap> SetPower(coll, 2, F.9*F.10);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
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);