#72266696

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#11390
commutator	C₂³×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂⁴×D₄	C₂³
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

2595 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=gj,b^a=bd,c^a=cg,c^b=ce,d^a=di,d^b=dj,d^c=dh,e^a=ehij,e^c=ei,e^d=ej,f^a=fj,g^b=gi,h^b=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266697, #72266721, #72266722
maximal normal subgroup of	#72266697, #72266721, #72266722
maximal quotient of	#72337721

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