#72340269

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

2843 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=gi,e^2=j,b^a=bi,c^a=cg,c^b=cf,d^a=dg,d^b=dh,d^c=di,e^a=efh,e^c=eh,e^d=ei,f^a=fi,f^c=fi,f^d=fj,g^b=gi,g^e=gj,h^a=hi,h^c=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340284, #72340307, #72340309, …
maximal normal subgroup of	#72340284, #72340307, #72340309, …

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