#72236729

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#342
commutator	C₂⁴	C₂²×C₄²
Frattini, cosocle	C₂³×D₄	C₂⁴
socle	C₂²	C₂²×#342
prime radical	C₂⁴×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂³×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5417 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,25)(20,24)(21,23)(22,26)(27,29)(28,30)(31,32);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,19,20,27,21,26,28,32)(18,22,24,29,23,25,30,31);(2,8)(6,14)(7,11)(13,15)(18,23)(19,26)(24,30)(27,32)

Polycyclic presentation:

2^10|a^2=g,b^2=i,c^2=e,e^2=j,c^a=cf,d^a=dij,d^c=dj,e^a=eh,f^a=fi,f^c=fh,f^e=fj,g^c=gi,h^c=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236751, #72240309, #72240321, …
maximal normal subgroup of	#72236751, #72240309, #72240321, …

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