#72263336

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#346446	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

3003 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=j,f^2=i,b^a=bg,c^a=cgj,c^b=chij,d^a=df,d^c=de,e^a=eh,e^b=ej,e^c=ei,f^a=fi,f^b=fi,f^c=fhij,f^d=fi,f^e=fj,g^d=gi,h^d=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263338
maximal normal subgroup of	#72263338

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