#72251850

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1510520
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3704 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=f,d^2=ij,g^2=j,h^2=j,b^a=bf,c^a=ch,c^b=cf,d^a=di,d^b=dghj,d^c=dhi,e^a=egj,e^c=eghi,e^d=eij,f^d=fj,g^e=gj,h^a=hj,h^c=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251866, #72251940, #72251959
maximal normal subgroup of	#72251866, #72251940, #72251959

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