#72321763

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	#7624084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2195 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=h,f^2=h,g^2=i,i^2=j,b^a=bgij,c^a=cg,c^b=ch,d^a=dj,d^b=dh,e^c=ehj,e^d=eh,f^a=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321949, #72322136, #72322260
maximal normal subgroup of	#72321949, #72322136, #72322260
maximal quotient of	#72343708, #72343733

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