#72318946

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	#2711373
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#2021	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2966 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)(5,9)(6,10)(11,23,25,17)(12,18,26,31)(14,27)(15,21)(16,22)(20,29)(24,30,28,32);(1,3)(2,7,10,19)(4,11)(5,12)(6,13)(8,17,21,23)(9,18,22,31)(14,28)(15,25)(16,26)(20,32,27,30)(24,29);(12,26)(18,31)(24,28)(30,32);(1,4)(2,8,10,21)(3,11)(5,14)(6,15)(7,17,19,23)(9,20,22,27)(12,24)(13,25)(16,29)(18,30,31,32)(26,28);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,20)(10,22)(11,24)(13,26)(15,29)(17,30)(19,31)(21,27)(23,32)(25,28)

Polycyclic presentation:

2^10|a^2=h,b^2=i,d^2=i,b^a=bg,c^b=cf,d^a=dfgij,d^b=df,e^a=eghi,e^b=eghi,e^c=eh,f^e=fj,g^a=gj,g^b=gj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319027, #72319065, #72319168, …
maximal normal subgroup of	#72319027, #72319065, #72319168, …

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