#72320267

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#22858
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#489284	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

999 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efhj,c^2=eh,d^2=eg,e^2=i,f^2=h,g^2=i,h^2=j,b^a=bf,c^a=cf,c^b=cfj,d^a=dg,d^b=dhj,d^c=dhi,e^b=ej,e^c=ej,f^a=fh,f^b=fj,f^c=fh,f^d=fj,g^a=gi,h^a=hj,h^c=hj

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320294, #72321057
maximal normal subgroup of	#72320294, #72321057

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