#72260652

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2029
commutator, Frattini, cosocle	C₂³×Q₈	C₂⁴
prime radical	#62811	C₂²
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

3049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=egj,e^2=j,f^2=j,b^a=bgh,c^a=ce,c^b=cefj,d^a=dhij,d^b=dhij,d^c=dhj,e^a=ei,e^b=eh,e^c=eij,e^d=ej,f^a=fhj,f^b=fhij,f^c=fhi,f^e=fj,g^c=gij,h^c=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260675, #72260762, #72260792, …
maximal normal subgroup of	#72260675, #72260762, #72260792, …

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