#72287601

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	#23001
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#33675	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8293 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

748 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)(3,11)(4,5)(6,9)(7,10)(8,14)(12,13)(15,16)(17,18,23,28)(19,24,22,26)(20,31,34,48)(21,35,38,44)(25,32,43,33)(27,30,45,37)(29,41,39,47)(36,42,46,40);(1,3,6,13,7,15,5,14)(2,8,4,16,10,12,9,11)(17,19,23,22)(18,24,28,26)(20,29,34,39)(21,30,38,37)(25,40,43,42)(27,35,45,44)(31,41,48,47)(32,36,33,46);(1,4,6,2,7,9,5,10)(3,12,13,16,15,8,14,11)(17,20,19,29,23,34,22,39)(18,25,26,42,28,43,24,40)(21,36,30,33,38,46,37,32)(27,41,44,31,45,47,35,48);(1,5,7,6)(3,13,15,14)(8,12)(11,16)(18,26)(19,22)(20,32,34,33)(21,30)(24,28)(25,41,43,47)(27,45)(29,46,39,36)(31,42,48,40)(37,38)

Polycyclic presentation:

2^10|a^2=j,b^2=gi,c^2=e,d^2=egh,e^2=i,f^2=hj,g^2=ij,h^2=j,b^a=bg,c^a=cf,c^b=cg,d^a=dgh,d^b=di,d^c=dfhi,e^a=ej,e^d=ej,f^a=fh,f^c=fhj,f^d=fhi,g^a=gij,g^c=gij,h^a=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

76 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.7*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5*F.7*F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, 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, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, 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);