#72284034

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

4001 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)(3,7)(4,12)(5,6)(8,13)(9,14)(10,11)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,29)(31,32);(1,3,6,11)(2,7,5,10)(4,9,13,15)(8,14,12,16)(23,28)(30,31);(1,4)(2,8,5,12)(3,9)(6,13)(7,14,10,16)(11,15)(17,19)(18,22,24,25)(20,26)(21,27)(23,30,28,31)(29,32);(9,15)(14,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32);(1,5,6,2)(3,10)(4,12,13,8)(7,11)(9,16)(14,15)(17,20)(18,23)(19,26,27,32)(21,29)(22,30,25,31)(24,28)

Polycyclic presentation:

2^10|b^2=ij,c^2=f,e^2=ghij,b^a=bg,c^a=ch,d^b=dg,d^c=dgh,e^a=eghij,e^b=efij,e^c=egi,e^d=ej,f^a=fi,f^d=fj,f^e=fij,g^c=gij,g^e=gj,h^b=hij,h^c=hi,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284100, #72284847, #72285190, …
maximal normal subgroup of	#72284100, #72284847, #72285190, …

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