#72341055

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	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233549	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3247 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=f,e^2=j,b^a=bi,c^a=cij,c^b=ch,d^a=dj,d^b=dfg,d^c=df,e^a=ej,e^b=ej,e^c=egij,e^d=egh,f^b=fi,f^e=fj,g^c=gj,g^d=gi,h^b=hj,h^c=hi,h^d=hij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341133
maximal normal subgroup of	#72341133

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