#72319071

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	#7623255	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2300 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=hj,e^2=f,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dhj,d^c=dg,e^a=eh,e^b=ehj,e^d=eij,f^d=fj,g^c=gi,g^d=gij,g^e=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319103, #72319316, #72319699
maximal normal subgroup of	#72319103, #72319316, #72319699

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