#72287643

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	#19128	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

894 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=efh,d^2=eghi,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bj,c^b=cf,d^a=dg,d^b=dfgj,d^c=dfgij,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	#72287651
maximal normal subgroup of	#72287651

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