#72321649

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#102
prime radical	#41×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4845 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)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,27)(20,31)(21,36)(22,24)(23,29)(25,34)(26,33)(28,43)(30,41)(32,39)(35,48)(37,44)(38,45)(40,46)(42,47);(1,3,5,11,6,12,9,15)(2,7,4,10,8,14,13,16)(17,19)(18,23,24,38)(20,32)(21,37)(22,28)(25,41,40,44)(26,42,36,39)(27,31,43,48)(29,46)(30,33)(34,45)(35,47);(2,8)(4,13)(7,14)(10,16);(1,4)(2,9)(3,10)(5,8)(6,13)(7,15)(11,14)(12,16)(18,24)(23,38)(25,40)(26,36)(27,43)(31,48)(39,42)(41,44);(2,8)(4,13)(7,14)(10,16)(18,24)(19,28)(25,40)(26,36)(29,45)(30,37)(31,48)(32,47)

Polycyclic presentation:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321717, #72322467
maximal normal subgroup of	#72321717, #72322467

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