#72281701

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊#95
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	C₂×C₂²⋊#95
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(17,18)(19,25)(20,24)(21,23)(22,27)(26,31)(28,30)(29,32);(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18)(19,25,27,22)(20,24)(21,23)(26,31,32,29)(28,30);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,20,26,21,27,28,32)(18,22,24,29,23,25,30,31);(18,23)(22,25)(24,30)(29,31)

Polycyclic presentation:

2^10|b^2=g,c^2=f,e^2=i,f^2=j,b^a=bh,c^a=cj,c^b=ce,d^a=dj,d^b=dj,e^a=ej,e^b=ej,f^b=fi,g^c=gij,h^c=hj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302147, #72302158, #72302165
maximal normal subgroup of	#72302147, #72302158, #72302165

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