#72303347

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#374038
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	#6437633	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

5338 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:

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

Polycyclic presentation:

2^10|b^2=hj,d^2=e,e^2=j,f^2=j,b^a=bh,c^b=cg,d^a=dj,d^b=df,d^c=dg,e^b=ej,e^c=ej,f^a=fi,f^b=fi,f^c=fj,g^d=gj,h^d=hij,h^f=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303363
maximal normal subgroup of	#72303363

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