#72321909

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1864
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#60673	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=ehij,e^2=i,f^2=h,g^2=i,h^2=j,b^a=bg,c^a=cf,c^b=cfhj,d^b=di,e^c=ej,f^a=fh,f^b=fhj,f^c=fh,g^a=gi,h^a=hj,h^b=hj,h^c=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321935
maximal normal subgroup of	#72321935

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