#72251676

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	#4894596
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂²×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4058 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)(4,17,19,32)(5,9)(6,10)(7,16)(8,20,26,27)(12,21)(13,28,18,31)(14,23)(15,24)(22,30,25,29);(1,3,12,15,6,16,5,14)(2,7,21,23,10,11,9,24)(4,18,20,29,19,13,27,30)(8,25,17,31,26,22,32,28);(1,4)(2,8,10,26)(3,13)(5,20)(6,19)(7,22,11,25)(9,17,21,32)(12,27)(14,30)(15,29)(16,18)(23,31,24,28);(7,11)(22,25)(23,24)(28,31);(3,14)(4,19)(5,12)(7,23)(9,21)(11,24)(13,29)(15,16)(17,32)(18,30)(22,31)(25,28)

Polycyclic presentation:

2^10|a^2=f,b^2=fgj,c^2=i,g^2=j,h^2=j,b^a=bfj,c^a=ch,c^b=cghj,d^a=dghj,d^b=di,e^a=eghj,e^b=egij,e^c=eij,f^c=fj,g^e=gj,h^b=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251679, #72251682, #72251715, …
maximal normal subgroup of	#72251679, #72251682, #72251715, …

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