#72251927

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	#2517181
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#449496	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=fghj,d^2=fgi,e^2=j,g^2=j,h^2=j,b^a=bi,c^a=cf,c^b=cfj,d^b=dfghi,d^c=dhi,e^a=ei,e^b=egi,e^c=ehi,e^d=ehij,f^b=fj,f^e=fj,g^c=gj,g^e=gj,h^b=hj,h^c=hj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251931, #72251940, #72251959
maximal normal subgroup of	#72251931, #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.7);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.7*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
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.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, 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);