#72316680

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7555
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=defghij,d^2=ij,f^2=j,g^2=ij,h^2=i,b^a=bg,c^a=cf,c^b=cd,d^a=di,e^a=ej,e^c=eij,f^a=fj,f^b=fi,f^c=fi,g^a=gij,g^b=gi,h^a=hi,h^b=hij,h^c=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316688, #72316855, #72316898
maximal normal subgroup of	#72316688, #72316855, #72316898

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