#72286014

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

2806 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)(3,7)(4,8,5,14)(6,9)(10,13)(11,16,12,15)(17,18)(19,23,29,34)(20,30)(21,25,33,39)(22,26)(24,32)(27,35,40,37)(28,36)(31,38,44,43)(41,48)(42,46)(45,47);(1,3)(2,7)(4,12)(5,11)(6,13)(8,16)(9,10)(14,15)(23,34)(25,39)(27,40)(28,42)(31,44)(36,46)(41,45)(47,48);(1,2)(3,10,13,7)(4,8,5,14)(6,9)(11,16)(12,15)(17,19)(18,23,26,34)(20,27)(21,28,33,42)(22,29)(24,37,30,35)(25,36)(31,45,44,41)(32,40)(38,47)(39,46)(43,48);(1,4,6,5)(2,8,9,14)(3,11)(7,15)(10,16)(12,13)(17,20)(18,24,26,30)(19,27,29,40)(21,31)(22,32)(23,35)(25,38,39,43)(28,41,42,45)(33,44)(34,37)(36,47)(46,48);(1,5,6,4)(2,8,9,14)(3,12)(7,16)(10,15)(11,13)(17,21)(18,25,26,39)(19,28,29,42)(20,31)(22,33)(23,36)(24,38,30,43)(27,41,40,45)(32,44)(34,46)(35,47)(37,48)

Polycyclic presentation:

2^10|a^2=f,c^2=fg,d^2=gi,e^2=gi,b^a=bf,c^a=cg,c^b=ch,d^a=dghi,d^b=df,d^c=dhij,e^a=eh,e^b=efh,e^c=egh,f^c=fj,f^d=fi,f^e=fij,g^a=gj,g^b=gi,g^c=gj,h^a=hij,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340258, #72340301, #72340329, …
maximal normal subgroup of	#72340258, #72340301, #72340329, …

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