#72341102

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
prime radical	#6432958	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5879 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)(3,8,4,9)(5,11)(6,10)(7,12)(13,15,14,16)(17,18)(19,23,29,36)(20,24)(21,25)(22,26)(27,35,30,45)(28,42,44,37)(31,38)(32,39)(33,46)(34,40)(41,48,43,47);(1,3)(2,8,12,9)(4,7)(5,13,6,14)(10,15)(11,16)(17,19,22,29)(18,23)(20,30)(21,28)(24,35,39,45)(25,37,40,42)(26,36)(27,32)(31,43,33,41)(34,44)(38,47)(46,48);(1,4,7,3)(2,9)(5,13)(6,14)(8,12)(10,15,11,16)(17,20,22,32)(18,24,26,39)(19,27)(21,33)(23,35)(25,38)(28,43,44,41)(29,30)(31,34)(36,45)(37,47,42,48)(40,46);(1,5,7,6)(2,10,12,11)(3,13,4,14)(8,15,9,16)(23,36)(27,30)(37,42)(41,43);(1,6,7,5)(2,11,12,10)(3,13,4,14)(8,15,9,16)(17,21,22,34)(18,25,26,40)(19,28,29,44)(20,31,32,33)(23,37,36,42)(24,38,39,46)(27,41,30,43)(35,47,45,48)

Polycyclic presentation:

2^10|a^2=g,b^2=fgij,c^2=fhi,d^2=j,e^2=i,b^a=bf,c^a=cfgh,c^b=cghi,d^a=dgj,d^b=dhj,d^c=dg,e^a=egj,e^b=egij,e^c=ehj,f^a=fi,f^b=fi,f^c=fi,f^d=fj,f^e=fi,g^b=gi,g^c=gj,h^a=hij,h^b=hj,h^c=hi

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341103, #72341145
maximal normal subgroup of	#72341103, #72341145

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