#72334837

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#4976232
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#417542	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3630 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)(3,7,13,21)(4,14)(5,17)(6,10)(8,16)(9,18)(11,25,27,22)(12,23,19,30)(15,28)(20,24)(26,31,29,32);(5,18)(7,21)(9,17)(12,19)(22,25)(31,32);(1,3)(2,7,10,21)(4,11)(5,19,18,12)(6,13)(8,22,14,25)(9,23)(15,29,20,26)(16,27)(17,30)(24,31)(28,32);(1,4)(2,8,10,14)(3,11)(5,20)(6,16)(7,22,21,25)(9,24,17,28)(12,26)(13,27)(15,18)(19,29)(23,32,30,31);(1,5)(2,9)(3,12)(4,15)(6,18)(7,23)(8,24)(10,17)(11,26)(13,19)(14,28)(16,20)(21,30)(22,31)(25,32)(27,29)

Polycyclic presentation:

2^10|a^2=h,c^2=f,d^2=i,b^a=bh,c^a=chi,c^b=ci,d^a=dg,d^b=dfi,d^c=dfi,e^a=eghj,e^b=egj,e^c=ehi,e^d=ehij,f^a=fj,f^e=fj,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	#72334842, #72334879, #72334880, …
maximal normal subgroup of	#72334842, #72334879, #72334880, …

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