#72251932

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	#5782713
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3574 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=fghj,d^2=fh,g^2=j,h^2=j,b^a=bf,c^a=ci,c^b=cfghj,d^a=df,d^b=dfghj,d^c=dhj,e^a=egi,e^c=eghij,e^d=eg,f^c=fj,f^d=fj,g^e=gj,h^a=hj,h^c=hj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251940, #72251956
maximal normal subgroup of	#72251940, #72251956

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