#72321120

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	#7623252	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17316 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2329 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=j,e^2=j,f^2=h,g^2=i,i^2=j,b^a=bgi,d^b=dh,d^c=dj,e^a=egj,e^c=ej,e^d=eh,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321406, #72322376, #72322909
maximal normal subgroup of	#72321406, #72322376, #72322909

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