#72263809

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
socle	C₂²	C₂×#2021
prime radical	#62773	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=hij,c^2=ei,d^2=egh,f^2=j,c^b=ce,d^b=defj,d^c=dfg,e^c=ej,e^d=ei,f^c=fh,f^d=fh,f^e=fj,g^b=ghij,g^c=ghi,g^d=gi,g^f=gj,h^b=hj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263814, #72316525
maximal normal subgroup of	#72263814, #72316525

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