#72341108

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

5729 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)(3,11,13,7)(4,8,5,9)(6,10)(12,15)(14,16)(17,18)(19,24,31,40)(20,32)(21,34)(22,25)(23,27)(26,37)(28,41,30,38)(29,39,35,46)(33,43)(36,42)(44,47,45,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,24,27,40)(20,28)(21,35,37,29)(22,30)(23,31)(25,41,32,38)(26,39)(33,44,36,45)(34,46)(42,47)(43,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,23,22)(18,25)(19,28,31,30)(21,36)(24,38)(26,43,34,42)(27,32)(29,45)(33,37)(35,44)(39,48,46,47)(40,41);(1,5,6,4)(2,9)(3,12)(7,16,11,15)(8,10)(13,14)(17,21)(18,26,27,34)(19,29)(20,33,22,36)(23,37)(24,39,40,46)(25,42)(28,44,30,45)(31,35)(32,43)(38,47)(41,48);(3,13)(4,5)(7,11)(8,9)(17,22,23,20)(18,25,27,32)(19,30,31,28)(21,36,37,33)(24,38,40,41)(26,43,34,42)(29,45,35,44)(39,48,46,47)

Polycyclic presentation:

2^10|a^2=g,b^2=h,c^2=hi,d^2=fgj,e^2=ij,b^a=bf,c^a=cfghj,c^b=ch,d^a=dfh,d^b=df,d^c=dfghij,e^b=ehj,e^c=ehj,e^d=ehi,f^a=fi,f^b=fij,f^c=fi,f^d=fj,f^e=fij,g^b=gi,g^c=gj,g^d=gj,h^a=hij,h^d=hij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341110, #72341169
maximal normal subgroup of	#72341110, #72341169

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