#72319984

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#121
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₂×C₄×#121
prime radical	#6467239	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fj,e^2=hj,f^2=i,h^2=j,c^a=cg,c^b=ci,d^a=dh,d^c=di,e^a=eh,e^d=ej,f^a=fj,g^c=gj,h^a=hj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320011, #72321186, C₂×#72319984
maximal normal subgroup of	#72320011, #72321186, C₂×#72319984
maximal quotient of	C₂×#72319984

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