#72321840

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	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2347 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)(3,7)(4,12)(5,9)(6,8)(10,16)(11,15)(13,14)(17,18,20,23,21,26,31,38,22,24,33,36,34,40,48,44)(19,27,29,45,30,37,47,25,28,42,46,43,35,41,32,39);(1,3,5,11,6,10,4,13)(2,7,9,15,8,16,12,14)(18,23,24,36)(19,28)(20,31)(21,34)(25,41,39,37)(26,44,40,38)(27,43,42,45)(29,32)(33,48)(46,47);(19,28)(25,39)(27,42)(29,46)(30,35)(32,47)(37,41)(43,45);(2,8)(3,10)(9,12)(11,13)(18,24)(23,36)(25,39)(26,40)(27,42)(37,41)(38,44)(43,45);(1,4,6,5)(2,9,8,12)(3,11,10,13)(7,14,16,15)(17,19,22,28)(18,25,24,39)(20,32,33,47)(21,35,34,30)(23,37,36,41)(26,45,40,43)(27,44,42,38)(29,31,46,48)

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=hj,f^2=h,g^2=i,i^2=j,b^a=bgij,d^a=dhj,d^b=dh,e^a=eghj,e^b=ehj,e^c=ej,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321949, #72322669
maximal normal subgroup of	#72321949, #72322669

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, 5, F.8*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*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
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);