#72250607

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#18203
commutator, Frattini, cosocle	C₂²×M₁₆	C₂⁴
prime radical	#6473339	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6,9,11)(7,15)(8,13,21,24)(10,14,23,25)(12,19)(16,26)(17,31,22,27)(18,28)(20,33)(29,36)(30,34)(32,35);(2,4)(6,11)(7,16,19,34)(8,17)(10,23)(12,26,15,30)(13,27)(18,20,35,36)(21,22)(24,31)(28,29,32,33);(5,7,17,16,10,20,8,18,9,19,22,34,23,36,21,35)(6,12,27,30,14,29,24,32,11,15,31,26,25,33,13,28);(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,26)(17,31)(18,28)(20,29)(21,24)(22,27)(23,25)(30,34)(32,35)(33,36)

Polycyclic presentation:

2^10|a^2=g,b^2=h,c^2=efij,f^2=i,i^2=j,b^a=bgh,c^a=ce,c^b=cefij,d^a=dhj,d^b=dhj,d^c=dhj,e^a=ej,e^b=eh,e^c=ehj,e^d=ej,f^a=fh,f^b=fhij,f^c=fhj,f^e=fj,g^c=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250622, #72256822, #72256826
maximal normal subgroup of	#72250622, #72256822, #72256826

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