#72287636

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623523	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fgh,e^2=ghi,f^2=h,g^2=i,h^2=j,i^2=j,d^a=dhi,d^b=dhi,d^c=dg,e^a=ej,e^b=ehi,e^c=eij,e^d=eghij,f^d=fi,f^e=fj,g^c=gj,g^d=gi,h^d=hj,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287659, #72318707, #72343439
maximal normal subgroup of	#72287659, #72318707, #72343439

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