#72339839

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#146565
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂×#532	C₂³
prime radical	#62815	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#117 ◁ #62815 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=g,c^2=e,d^2=j,f^2=j,g^2=ij,h^2=j,c^a=cf,c^b=cd,d^a=di,d^b=dh,d^c=di,e^b=eij,e^d=ej,f^a=fj,f^b=fij,f^c=fj,f^d=fj,g^c=gh,g^d=gij,h^b=hi,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339845
maximal normal subgroup of	#72339845

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