#72310880

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

2805 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=gi,e^2=fgj,b^a=bi,c^a=cg,c^b=cf,d^a=di,d^b=dgh,d^c=dij,e^a=eh,e^b=efghj,e^c=eh,e^d=efj,f^a=fi,f^c=fi,f^d=fj,f^e=fj,g^b=gi,g^e=gj,h^b=hi,h^c=hj,h^d=hi,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310940, #72310982, #72340248, …
maximal normal subgroup of	#72310940, #72310982, #72340248, …

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