#72340485

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	#35346	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2859 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)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(19,27)(24,39)(30,34)(40,46);(1,2)(3,9)(4,7,14,12)(5,8)(6,11)(10,15,13,16)(17,18)(19,28,27,23)(20,24,32,39)(21,33)(22,26)(25,36)(29,37)(30,38,34,41)(31,46,35,40)(42,43)(44,48)(45,47);(1,3)(2,6,8,9)(4,13)(5,11)(7,15,12,16)(10,14)(17,19)(18,23)(20,29)(21,34,36,30)(22,27)(24,37)(25,41,33,38)(26,28)(31,44,35,45)(32,42)(39,43)(40,47,46,48);(1,4)(2,7,8,12)(3,10)(5,14)(6,15,9,16)(11,13)(17,20,22,32)(18,24,26,39)(19,29,27,42)(21,35)(23,37,28,43)(25,40)(30,45)(31,36)(33,46)(34,44)(38,47)(41,48);(4,14)(7,12)(17,21)(18,25)(19,30)(20,31)(22,36)(23,38)(24,40)(26,33)(27,34)(28,41)(29,44)(32,35)(37,47)(39,46)(42,45)(43,48)

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=gj,b^a=bf,c^a=ch,c^b=chj,d^a=dfhj,d^b=dfgh,d^c=dgij,e^b=ej,e^c=ef,e^d=ehi,f^c=fj,f^d=fi,g^a=gij,g^b=gij,g^e=gj,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340495, #72341110, #72341124
maximal normal subgroup of	#72340495, #72341110, #72341124

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