#72251922

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	#1965979
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3776 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=fgj,d^2=fh,h^2=j,b^a=bi,c^a=cf,c^b=cfhj,d^a=di,d^b=df,e^a=ei,e^b=eg,e^c=ehj,e^d=ehi,f^b=fj,f^d=fj,f^e=fj,g^d=gj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251931, #72251940, #72251951
maximal normal subgroup of	#72251931, #72251940, #72251951

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