#72321750

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1853
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×#41	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1362 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=eghj,f^2=h,g^2=i,h^2=j,b^a=bg,c^b=cf,d^b=di,e^a=ei,e^c=ej,f^b=fh,f^c=fh,g^a=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321818, #72322289
maximal normal subgroup of	#72321818, #72322289

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