#72340291

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

2843 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=f,c^2=g,d^2=fg,e^2=gi,b^a=bg,c^a=cgi,c^b=cfg,d^a=dh,d^b=dhj,d^c=dfghj,e^b=ehi,e^c=egij,e^d=efij,f^a=fi,f^c=fi,f^d=fj,f^e=fj,g^b=gi,g^d=gj,h^b=hj,h^c=hij,h^d=hi,h^e=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340293, #72340295, #72340309, …
maximal normal subgroup of	#72340293, #72340295, #72340309, …

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