#72334820

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3067826
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#429959	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3380 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=f,d^2=ij,b^a=bgh,c^a=ci,d^a=dghij,d^b=dfij,d^c=df,e^a=eh,e^b=egh,e^c=ehij,e^d=ehj,f^e=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334823, #72334852, #72334856, …
maximal normal subgroup of	#72334823, #72334852, #72334856, …

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