#72334854

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1907613
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#454968	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3070 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)(3,7,13,19)(4,8,15,24)(5,9)(6,10)(11,22)(12,21,16,25)(14,23,17,26)(18,27)(20,29)(28,32)(30,31);(7,19)(14,17)(20,22)(21,25)(23,26)(28,30);(1,3)(2,7,10,19)(4,11)(5,16,18,12)(6,13)(8,22,24,20)(9,25)(14,30,17,28)(15,29)(21,27)(23,31)(26,32);(1,4)(2,8,10,24)(3,11)(5,17)(6,15)(7,20,19,22)(9,26,27,23)(12,28)(13,29)(14,18)(16,30)(21,31,25,32);(1,5)(2,9)(3,12)(4,14)(6,18)(7,21)(8,23)(10,27)(11,28)(13,16)(15,17)(19,25)(20,31)(22,32)(24,26)(29,30)

Polycyclic presentation:

2^10|a^2=g,c^2=f,d^2=i,b^a=bh,c^a=chi,c^b=ci,d^a=dgi,d^b=dfi,d^c=df,e^b=egh,e^c=eh,e^d=ehj,f^a=fj,f^e=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334856, #72334872, #72334892, …
maximal normal subgroup of	#72334856, #72334872, #72334892, …

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