#72237893

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	C₂	#2109305
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=j,c^2=h,e^2=h,b^a=bg,c^a=ch,c^b=cf,d^a=dj,d^b=di,d^c=dgij,e^a=eh,e^b=egi,e^c=eh,e^d=eij,f^c=fj,f^d=fj,f^e=fj,g^c=gj,g^e=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237907, #72240477, #72251069, …
maximal normal subgroup of	#72237907, #72240477, #72251069, …

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