#72287103

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂×#1580
prime radical	C₂×#508	C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11411 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=e,c^2=eg,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bij,c^b=ci,d^a=dh,d^c=df,e^d=ej,f^a=fj,f^c=fh,h^c=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287139, #72287361, #72287377, …
maximal normal subgroup of	#72287139, #72287361, #72287377, …
maximal quotient of	#72323198

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