#72322280

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×#1864
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efhj,c^2=ehi,e^2=i,f^2=h,g^2=i,h^2=j,b^a=bf,c^a=cfgi,c^b=cfgj,d^a=di,d^b=di,e^b=ej,e^c=ej,f^a=fh,f^b=fj,f^c=fhi,g^a=gi,h^a=hj,h^c=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322289
maximal normal subgroup of	#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, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.6*F.8*F.10);;
gap> SetPower(coll, 3, F.5*F.8*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, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
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.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);