#72338917

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#5593964
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#394262	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3075 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gi,c^2=j,e^2=gij,f^2=j,b^a=bf,c^a=cf,c^b=ch,d^a=dg,d^b=dfgh,d^c=dhij,e^a=eg,e^b=egij,e^c=ehi,e^d=egij,f^b=fj,f^c=fj,f^d=fj,f^e=fj,g^b=gj,g^c=gj,h^b=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338928
maximal normal subgroup of	#72338928

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