#72237867

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#486091
commutator, Frattini, cosocle	#381	C₂⁴
LCS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ #2308 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=hi,e^2=hj,f^2=hi,b^a=bf,c^b=ce,d^a=dhi,d^b=dhi,d^c=dh,e^a=eg,e^b=eh,e^c=ehj,e^d=ej,f^a=fhi,f^b=fhi,f^c=fgij,f^e=fi,g^b=gi,g^c=gj,g^e=gj,g^f=gj,h^a=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237869
maximal normal subgroup of	#72237869

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