#72287630

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	#23029
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#490656	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

773 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=i,d^2=gi,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bj,c^a=cg,c^b=cgj,d^a=dg,d^b=df,d^c=dfhj,e^d=ej,f^a=fij,f^b=fh,f^c=fhj,f^d=fhi,g^a=gij,g^b=gij,g^c=gij,h^b=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287651, #72319994, #72320455, …
maximal normal subgroup of	#72287651, #72319994, #72320455, …

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