#72284767

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

2845 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)(3,4)(5,14)(6,10)(7,8)(9,16)(11,13)(12,15)(18,23)(19,27)(21,34)(25,35)(37,39)(38,41)(40,46)(47,48);(1,3)(2,7)(4,10)(5,11,16,12)(6,8)(9,13,14,15)(17,18)(19,28)(20,25,33,35)(21,26)(22,23)(24,27)(29,37,31,39)(30,38)(32,40,45,46)(34,36)(41,42)(43,48,44,47);(1,4,6,7)(2,8,10,3)(5,15)(9,11)(12,14)(13,16)(17,19,22,27)(18,24,23,28)(20,31)(21,30,34,42)(25,39)(26,41,36,38)(29,33)(32,43)(35,37)(40,48)(44,45)(46,47);(1,5,6,16)(2,9)(3,11)(4,13,7,15)(8,12)(10,14)(17,20)(18,25,23,35)(19,29,27,31)(21,32)(22,33)(24,37)(26,40,36,46)(28,39)(30,43,42,44)(34,45)(38,47)(41,48);(1,5)(2,9,10,14)(3,12,8,11)(4,13)(6,16)(7,15)(17,21)(18,26,23,36)(19,30,27,42)(20,32)(22,34)(24,38)(25,40,35,46)(28,41)(29,43,31,44)(33,45)(37,47)(39,48)

Polycyclic presentation:

2^10|b^2=f,c^2=fi,d^2=gij,e^2=g,b^a=bi,c^a=cfi,c^b=cfi,d^a=dfh,d^b=dfg,d^c=dhj,e^a=efj,e^b=efhi,e^c=eghj,e^d=eij,f^d=fi,f^e=fij,g^a=gij,g^b=gi,g^c=gj,h^b=hij,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190, #72286044, #72340228, …
maximal normal subgroup of	#72285190, #72286044, #72340228, …

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