#72341077

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5919 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)(3,6)(4,12)(5,9)(7,11)(8,14)(10,15)(13,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,32)(27,38)(28,44)(29,37)(31,39)(33,36)(34,41)(35,42)(40,46)(43,48)(45,47);(1,3)(2,6)(4,13,14,10)(5,11)(7,9)(8,16,12,15)(17,19)(18,23)(20,27,32,42)(21,33)(22,29)(24,38,30,35)(25,36)(26,37)(28,34)(31,43,46,45)(39,47,40,48)(41,44);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(24,30)(27,42)(39,40)(43,45);(1,3)(2,7)(4,13)(5,11)(6,9)(8,15)(10,14)(12,16)(17,20,22,32)(18,24)(19,27)(21,31,34,46)(23,35,37,38)(25,40)(26,30)(28,45)(29,42)(33,43)(36,47,44,48)(39,41);(1,4,5,14)(2,8,9,12)(3,10,11,13)(6,15,7,16)(17,21,22,34)(18,25)(19,28)(20,31,32,46)(23,36,37,44)(24,39)(26,41)(27,43)(29,33)(30,40)(35,47,38,48)(42,45)

Polycyclic presentation:

2^10|b^2=g,d^2=fgj,e^2=fgij,b^a=bg,c^a=cg,c^b=cg,d^a=dgi,d^b=df,d^c=dfg,e^a=egh,e^b=efhij,e^d=efgi,f^a=fj,f^b=fj,f^d=fj,f^e=fi,g^d=gj,g^e=gi,h^b=hij,h^c=hi,h^d=hi,h^e=hij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341091, #72341103, …
maximal normal subgroup of	#72341083, #72341091, #72341103, …

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