#72337585

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

2851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=j,b^a=bf,c^b=cij,d^a=df,d^b=dgh,d^c=dfij,e^a=eh,e^c=egj,f^b=fi,f^e=fj,g^a=gi,g^d=gj,h^b=hj,h^c=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337590, #72340457, #72340494
maximal normal subgroup of	#72337590, #72340457, #72340494

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