#72284428

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
prime radical	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2911 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)(4,7)(5,8)(6,11)(10,14)(12,13)(15,16)(17,18)(19,27)(20,31)(21,35)(22,26)(23,30)(24,33)(25,36)(28,41)(29,38)(32,37)(34,44)(39,46)(40,43)(42,48)(45,47);(1,3)(2,6,8,9)(4,10,13,16)(5,11)(7,15)(12,14)(17,19)(18,23,26,27)(20,32)(21,29,36,43)(22,30)(24,37,31,41)(25,40)(28,33)(34,45,46,42)(35,38)(39,48)(44,47);(1,2,5,8)(3,6)(4,12,13,7)(9,11)(10,14)(15,16)(17,20,22,33)(18,24,26,31)(19,28)(21,34)(23,37)(25,39)(27,41)(29,42,43,45)(30,32)(35,44)(36,46)(38,47,40,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(20,33)(21,36)(24,31)(25,35)(34,46)(39,44)(42,45)(47,48);(1,4)(2,7,8,12)(3,10,11,16)(5,13)(6,14)(9,15)(17,21,22,36)(18,25)(19,29,30,43)(20,34)(23,38)(24,39,31,44)(26,35)(27,40)(28,42)(32,45)(33,46)(37,47,41,48)

Polycyclic presentation:

2^10|b^2=fg,c^2=fi,e^2=ghij,b^a=bf,c^a=cfi,c^b=cgh,d^b=dfh,d^c=dfi,e^a=efhj,e^b=egij,e^c=efh,e^d=ehi,f^b=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gj,g^d=gj,g^e=gj,h^b=hij,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284848, #72285772, #72310940, …
maximal normal subgroup of	#72284848, #72285772, #72310940, …

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