#72284633

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

3482 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)(3,10)(4,7)(5,13)(6,9)(8,16)(11,12)(14,15)(17,18)(19,23)(20,28)(21,29)(22,26)(24,31)(25,32)(27,30);(1,3,6,7)(2,4,9,10)(5,14,16,11)(8,15,13,12)(17,19)(18,23)(20,29)(21,28)(22,27)(24,32)(25,31)(26,30);(1,4)(2,7)(3,9)(5,15)(6,10)(8,14)(11,13)(12,16)(24,28)(25,29);(1,5)(2,8,9,13)(3,11)(4,12,10,15)(6,16)(7,14)(17,20)(18,24,26,28)(19,25)(21,30,32,23)(22,31)(27,29);(2,9)(3,7)(4,10)(5,16)(11,14)(12,15)(17,21)(18,25)(19,24)(20,30)(22,32)(23,31)(26,29)(27,28)

Polycyclic presentation:

2^10|b^2=ij,d^2=gh,b^a=bij,c^a=cf,c^b=cf,d^a=df,d^b=di,d^c=dfg,e^a=ehi,e^b=efghi,e^c=efgij,e^d=ehi,f^d=fi,f^e=fj,g^a=gi,g^b=gi,g^d=gj,g^e=gj,h^b=hi,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190, #72310874, #72340239, …
maximal normal subgroup of	#72285190, #72310874, #72340239, …

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