#72287484

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#102
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#6358421	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#135 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1283 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=g,c^2=e,d^2=efh,e^2=hi,f^2=hi,g^2=i,h^2=j,b^a=bg,c^a=cg,c^b=cgi,d^a=dgi,d^c=df,e^d=eh,f^a=fi,f^c=fij,f^d=fh,g^a=gi,g^c=gi,h^d=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287545
maximal normal subgroup of	#72287545

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