#72284649

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgh,e^2=g,b^a=bh,c^a=chij,c^b=cg,d^b=dgi,d^c=dghj,e^a=eg,e^b=eghi,e^c=efgj,e^d=ej,f^a=fi,f^c=fj,f^d=fj,f^e=fij,g^b=gj,g^c=gij,h^b=hi,h^c=hi,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285163, #72285526, #72311043, …
maximal normal subgroup of	#72285163, #72285526, #72311043, …

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