#72338236

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7555
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=ij,c^2=dfghj,d^2=ij,e^2=i,f^2=j,b^a=be,c^a=cd,c^b=cefj,d^a=di,e^a=ei,e^b=ei,e^c=ej,f^a=fij,f^b=fj,f^c=fij,g^c=gj,h^a=hi,h^c=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338240, #72338250, #72338258
maximal normal subgroup of	#72338240, #72338250, #72338258

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