#72283904

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

2853 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,15)(13,16)(23,36)(24,30)(25,34)(27,31)(28,43)(37,45)(38,40)(47,48);(1,3)(2,7)(4,11,5,13)(6,12)(8,14,9,16)(10,15)(17,18)(19,23,29,36)(20,30,33,24)(21,34,35,25)(22,26)(27,37)(28,38)(31,45)(32,39)(40,43)(41,46)(42,47,44,48);(1,4,6,5)(2,8,10,9)(3,11)(7,14)(12,13)(15,16)(17,19)(18,23,26,36)(20,31,33,27)(21,28,35,43)(22,29)(24,37)(25,40)(30,45)(32,44)(34,38)(39,48,41,47)(42,46);(3,12)(8,9)(11,13)(14,16)(17,20,22,33)(18,24,26,30)(19,27,29,31)(21,32,35,46)(23,37,36,45)(25,41,34,39)(28,44,43,42)(38,47,40,48);(1,5,6,4)(2,9)(3,11)(7,14,15,16)(8,10)(12,13)(17,21,22,35)(18,25,26,34)(19,28)(20,32)(23,38)(24,39)(27,42,31,44)(29,43)(30,41)(33,46)(36,40)(37,47,45,48)

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=j,e^2=ghi,b^a=bf,c^a=cg,c^b=cfgij,d^a=dfgij,d^b=dh,d^c=dhij,e^a=efgij,e^b=efg,e^c=ehij,e^d=ehi,f^c=fi,f^d=fj,f^e=fi,g^b=gi,g^d=gj,g^e=gj,h^a=hij,h^b=hj,h^c=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283944, #72284072, #72285190, …
maximal normal subgroup of	#72283944, #72284072, #72285190, …

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