#72320382

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#22858
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#489287	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ei,d^2=egi,e^2=i,f^2=h,g^2=i,h^2=j,c^a=ch,c^b=cf,d^a=dg,d^b=di,d^c=dhij,e^c=ej,f^a=fj,f^b=fh,f^c=fhj,f^d=fj,g^a=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320455, #72320730
maximal normal subgroup of	#72320455, #72320730

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