#72321003

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	#490676	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)(3,6)(4,13)(5,7)(8,14)(9,11)(10,16)(12,15)(17,18)(19,24)(20,25)(21,27)(22,26)(23,28)(29,36)(30,38)(31,37)(32,39)(33,41)(34,40)(35,42)(43,47)(44,46)(45,48);(2,7,3,10)(4,11)(5,15)(8,13,12,9)(14,16)(17,19,21,30,23,32,22,31)(18,24,27,38,28,39,26,37)(20,29,34,44,33,43,35,45)(25,36,40,46,41,47,42,48);(1,3,11,13,6,2,4,9)(5,7,15,12,16,10,14,8)(17,20,22,35,23,33,21,34)(18,25,26,42,28,41,27,40)(19,29,31,45,32,43,30,44)(24,36,37,48,39,47,38,46);(1,4,6,11)(2,3)(5,15,16,14)(9,13)(18,26,28,27)(20,33)(24,37,39,38)(25,40,41,42)(29,43)(34,35)(36,46,47,48)(44,45)

Polycyclic presentation:

2^10|b^2=e,c^2=ehij,d^2=egh,e^2=i,f^2=hj,g^2=ij,h^2=j,b^a=bf,c^a=ch,c^b=cfh,d^a=dg,d^b=dj,d^c=dhi,e^a=ej,e^c=ej,f^a=fh,f^b=fhj,f^c=fhj,f^d=fj,g^a=gij,g^b=gj,g^c=gj,h^a=hj,h^b=hj,h^c=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321033, #72321057
maximal normal subgroup of	#72321033, #72321057

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