#72321989

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	#33526	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17316 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2426 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)(4,13)(5,8)(6,9)(10,16)(11,14)(12,15)(17,18,21,25,20,27,32,42,22,26,35,36,33,43,48,45)(19,24,30,37,31,41,34,44,29,40,47,23,28,38,46,39);(1,3,5,11,6,12,4,10)(2,7,8,14,9,15,13,16)(17,19)(18,23,26,37)(20,28)(21,34)(22,29)(24,36,40,25)(27,44,43,39)(30,32)(31,33)(35,46)(38,45,41,42)(47,48);(17,19)(18,24)(20,31)(21,30)(22,29)(23,36)(25,37)(26,40)(27,41)(28,33)(32,34)(35,47)(38,43)(39,45)(42,44)(46,48);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,20)(18,25)(19,28)(22,33)(23,24)(26,36)(27,45)(29,31)(30,47)(32,48)(37,40)(38,39)(41,44)(42,43);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(18,26)(19,29)(25,36)(27,43)(28,31)(30,47)(34,46)(42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322136, #72322669, #72343989
maximal normal subgroup of	#72322136, #72322669, #72343989
maximal quotient of	#72343627, #72343681

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