#72310967

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2770 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)(3,5)(4,8)(6,10)(7,9)(11,14)(12,15)(13,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,32)(27,36)(28,37)(29,35)(31,39)(33,40)(34,41)(38,42)(43,48)(44,45)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,12)(13,15)(14,16)(17,19)(18,23)(20,27)(21,33)(22,29)(24,38)(25,40)(26,35)(28,34)(30,36)(31,43)(32,42)(37,41)(39,48)(44,47)(45,46);(1,4,6,13)(2,8,10,16)(3,11,9,15)(5,12,7,14)(21,34)(23,35)(27,42)(39,44);(1,5,6,7)(2,9)(3,10)(4,12)(8,15,16,11)(13,14)(17,20,22,32)(18,24)(19,27)(21,31,34,45)(23,36,35,38)(25,39)(26,30)(28,43)(29,42)(33,46)(37,47,40,48)(41,44);(2,10)(4,13)(5,7)(8,16)(17,21)(18,25)(19,28)(20,31)(22,34)(23,37)(24,39)(26,41)(27,43)(29,33)(30,44)(32,45)(35,40)(36,47)(38,48)(42,46)

Polycyclic presentation:

2^10|c^2=ij,d^2=ghi,b^a=bij,c^a=cf,c^b=cfg,d^a=dfh,d^b=dfgi,d^c=dfij,e^a=ehij,e^b=ehj,e^c=egi,e^d=eij,f^b=fi,f^d=fij,f^e=fj,g^a=gi,g^b=gi,g^d=gj,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311029
maximal normal subgroup of	#72311029

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