#72321672

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1762
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1571 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=ei,e^2=i,f^2=h,g^2=i,h^2=j,b^a=bf,c^b=chij,d^b=di,e^b=ej,f^a=fh,f^b=fhij,f^c=fj,g^a=gi,h^a=hj,h^b=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321708
maximal normal subgroup of	#72321708

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