#72251891

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3853977
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4629 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fhj,d^2=ij,h^2=j,b^a=bf,c^a=ci,c^b=cij,d^a=di,d^b=dh,d^c=dg,e^b=egi,e^c=eghj,e^d=ei,f^c=fj,f^d=fj,g^b=gj,g^c=gj,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251900, #72251940, #72251959
maximal normal subgroup of	#72251900, #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, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.8*F.10);;
gap> SetPower(coll, 4, F.9*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.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, 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, 3, 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);