#72234164

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14679
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂⁶	C₂⁴
prime radical	C₂×D₄×H(5,𝔽₂)	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

33756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

(9,10)(11,15)(12,14)(13,16);(9,11,12,16)(10,13,14,15);(1,2)(3,7,8,5)(4,6)(13,15);(1,3,4,8)(2,5,6,7)

Polycyclic presentation:

2^10|b^2=j,c^2=e,d^2=i,b^a=bj,c^a=cg,c^b=ch,d^c=def,e^d=ei,f^c=fi,f^d=fi,g^b=gj,h^a=hj

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234173, #72234200, #72258972, …
maximal normal subgroup of	#72234173, #72234200, #72258972, …
maximal quotient of	#72234165, #72234200, #72234201, …

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