#72335087

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	#6296722
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#464740	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2949 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)(4,15,16,8)(5,9)(6,10)(7,12)(13,20,26,24)(14,21)(17,22,18,30)(19,23)(25,27)(28,31,29,32);(3,12)(4,16)(8,15)(14,27)(20,24)(28,29);(1,3)(2,7,10,11)(4,13)(5,14,19,27)(6,12)(8,20,15,24)(9,21)(16,26)(17,28,18,29)(22,31)(23,25)(30,32);(1,4,6,16)(2,8)(3,13,12,26)(5,18,19,17)(7,20)(9,22)(10,15)(11,24)(14,29,27,28)(21,31)(23,30)(25,32);(1,5)(2,9)(3,14)(4,17)(6,19)(7,21)(8,22)(10,23)(11,25)(12,27)(13,28)(15,30)(16,18)(20,31)(24,32)(26,29)

Polycyclic presentation:

2^10|a^2=g,c^2=fhij,d^2=i,b^a=bh,c^a=cf,c^b=ci,d^a=dgi,d^b=dfhj,e^a=egh,e^b=eg,e^c=ehj,e^d=ei,f^c=fj,f^e=fj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335088, #72335094, #72335107, …
maximal normal subgroup of	#72335088, #72335094, #72335107, …

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