#72341207

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	#35527	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2723 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)(3,7,5,9)(4,12)(6,10)(8,14)(11,15,13,16)(17,18)(19,23)(20,24)(21,26,33,39)(22,25)(27,34)(28,36)(29,43,42,35)(30,45)(31,37,46,40)(32,38)(41,48,44,47);(1,3,6,5)(2,7)(4,13,14,11)(8,15)(9,10)(12,16)(17,19,22,28)(18,23,25,36)(20,30)(21,29,33,42)(24,34)(26,35,39,43)(27,32)(31,41)(37,48)(38,45)(40,47)(44,46);(1,4)(2,8)(3,11,5,13)(6,14)(7,15,9,16)(10,12)(17,20,22,32)(18,24,25,38)(19,27,28,30)(21,31)(23,34,36,45)(26,40)(29,44)(33,46)(35,47)(37,39)(41,42)(43,48);(1,3,6,5)(2,7,10,9)(4,13,14,11)(8,15,12,16)(18,25)(19,28)(26,39)(29,42);(1,5,6,3)(2,9,10,7)(4,11,14,13)(8,15,12,16)(17,21,22,33)(18,26,25,39)(19,29,28,42)(20,31,32,46)(23,35,36,43)(24,37,38,40)(27,41,30,44)(34,47,45,48)

Polycyclic presentation:

2^10|a^2=f,b^2=gi,c^2=fij,d^2=j,e^2=i,b^a=bfg,c^a=cfj,c^b=ch,d^a=dgi,d^b=dgij,d^c=dh,e^a=ehi,e^b=eg,e^c=eghij,f^b=fj,f^d=fj,f^e=fi,g^a=gj,g^c=gij,h^a=hi,h^b=hij,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341233, #72341244
maximal normal subgroup of	#72341233, #72341244

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