#72285341

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2877 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)(3,9,13,7)(4,5,15,16)(6,8)(10,11)(12,14)(17,18)(19,27,30,23)(20,24,33,39)(21,34)(22,26)(25,35)(28,40)(29,41,43,37)(31,36)(32,38,46,45)(42,47)(44,48);(1,3,6,13)(2,7,8,9)(4,14,15,10)(5,12,16,11)(17,19)(18,23)(20,31)(21,29)(22,30)(24,36)(25,37)(26,27)(28,33)(32,44)(34,41)(35,43)(38,47)(39,40)(42,46)(45,48);(1,4)(2,5,8,16)(3,10)(6,15)(7,11,9,12)(13,14)(17,18)(19,27)(20,24)(21,25,35,34)(22,26)(23,30)(28,36)(29,41,43,37)(31,40)(32,45,46,38)(33,39)(42,48,44,47);(1,5,6,16)(2,4,8,15)(3,11)(7,10)(9,14)(12,13)(17,20)(18,24,26,39)(19,28,30,31)(21,32)(22,33)(23,36)(25,38,34,45)(27,40)(29,42,43,44)(35,46)(37,47)(41,48);(1,5)(2,4)(3,12,13,11)(6,16)(7,10,9,14)(8,15)(17,21)(18,25,26,34)(19,29,30,43)(20,32)(22,35)(23,37)(24,38,39,45)(27,41)(28,42,31,44)(33,46)(36,47)(40,48)

Polycyclic presentation:

2^10|a^2=f,b^2=ij,c^2=fh,d^2=ghij,e^2=gh,b^a=bg,c^a=ch,c^b=ci,d^a=dg,d^b=dfgh,d^c=dhi,e^a=eh,e^b=efhi,e^c=eg,e^d=eij,f^b=fi,f^c=fij,f^d=fi,f^e=fij,g^a=gi,g^c=gi,g^d=gj,g^e=gj,h^a=hij,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526, #72310969, #72310975, …
maximal normal subgroup of	#72285526, #72310969, #72310975, …

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