#72287259

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁵
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#135 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=ij,b^2=g,c^2=ij,d^2=gij,f^2=hj,g^2=i,h^2=j,b^a=bg,c^a=ci,c^b=cgi,d^a=dgi,d^b=di,d^c=df,e^a=ei,e^c=eij,f^a=fi,f^c=fh,f^d=fhi,g^a=gi,g^c=gi,h^c=hj,h^d=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287387, #72287545
maximal normal subgroup of	#72287387, #72287545

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