#72233595

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1898
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂²×D₄²	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4921 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:

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

Polycyclic presentation:

2^10|a^2=hij,b^2=e,e^2=j,b^a=bf,c^a=ch,c^b=chij,d^a=dg,d^b=dhj,e^a=eh,e^c=ej,e^d=ej,f^a=fj,f^b=fh,f^c=fj,f^d=fi,f^e=fj,g^b=gi,h^b=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233605, #72233614, #72233619
maximal normal subgroup of	#72233605, #72233614, #72233619

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