#72284312

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

3480 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,14)(15,16)(23,32)(25,31);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,18)(19,23)(20,29)(21,31)(22,26)(24,30)(25,27)(28,32);(1,4,5,12)(2,7)(3,9,10,15)(6,13)(8,11)(14,16)(17,19)(18,23)(20,27,30,21)(22,28)(24,31,29,25)(26,32);(9,15)(13,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,25)(26,29)(31,32);(4,12)(6,14)(17,21,22,27)(18,25,26,31)(19,20,28,30)(23,29,32,24)

Polycyclic presentation:

2^10|b^2=i,c^2=f,e^2=j,b^a=bg,c^a=ch,d^b=dgj,d^c=dghj,e^a=eh,e^b=efgij,e^c=efghj,f^a=fi,f^d=fj,f^e=fj,g^c=gij,g^e=gj,h^b=hij,h^c=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285163, #72285190, #72285517, …
maximal normal subgroup of	#72285163, #72285190, #72285517, …

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