#72340488

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	#245228	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2730 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)(5,9)(6,10)(11,12)(13,15)(14,16)(20,30)(25,39)(27,40)(35,42);(1,3,6,8)(2,7,10,4)(5,11)(9,13)(12,16)(14,15)(17,18)(19,23)(20,31,30,24)(21,25,33,39)(22,26)(27,37,40,34)(28,42,43,35)(29,36)(32,38)(41,47)(44,45)(46,48);(1,4,6,7)(2,8,10,3)(5,13)(9,15)(11,16)(12,14)(17,19,22,29)(18,23,26,36)(20,27)(21,28)(24,37)(25,35)(30,40)(31,34)(32,41,44,48)(33,43)(38,47,45,46)(39,42);(1,5)(2,9,10,16)(3,11,8,15)(4,12)(6,14)(7,13)(17,20,22,30)(18,24,26,31)(19,27)(21,32,33,44)(23,34)(25,38,39,45)(28,41)(29,40)(35,46)(36,37)(42,47)(43,48);(2,10)(3,8)(4,7)(5,14)(17,21)(18,25)(19,28)(20,32)(22,33)(23,35)(24,38)(26,39)(27,41)(29,43)(30,44)(31,45)(34,46)(36,42)(37,47)(40,48)

Polycyclic presentation:

2^10|b^2=fij,c^2=fi,d^2=ghij,b^a=bf,c^b=cf,d^a=dfh,d^b=dfhi,d^c=dg,e^a=ehj,e^b=egi,e^c=egh,e^d=eij,f^d=fi,f^e=fj,g^a=gij,g^b=gj,g^c=gi,g^d=gj,h^b=hi,h^c=hij,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340495
maximal normal subgroup of	#72340495

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