#72287451

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	#489290	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

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=efhj,d^2=egh,e^2=i,f^2=h,g^2=ij,h^2=j,c^b=cf,d^a=dg,d^b=dfgi,d^c=dfgi,e^c=ej,e^d=ej,f^b=fh,f^c=fj,f^d=fh,g^a=gij,g^b=gij,g^c=gij,h^b=hj,h^d=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287455
maximal normal subgroup of	#72287455

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