#72287592

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	#489286	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

907 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=i,c^2=e,d^2=eghj,e^2=i,f^2=hj,g^2=ij,h^2=j,c^b=cf,d^a=dg,d^b=dhij,d^c=dfgh,e^b=ej,e^d=ej,f^b=fh,f^c=fhj,f^d=fhj,g^a=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
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.9);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.10);;
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, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, 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.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
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, 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);