#72250331

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	#18152
commutator, Frattini, cosocle	C₂²×M₁₆	C₂⁴
prime radical	#142258	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1975 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)(5,6,10,15,25,16,28,36,11,13,32,34,8,17,24,27)(7,18,20,12,19,30,14,31,22,23,35,33,21,9,26,29);(2,4)(5,7,11,22)(6,12)(8,19,25,21)(9,27)(10,26,32,14)(13,33)(15,23)(16,29)(17,31)(18,34)(20,28,35,24)(30,36);(5,8,11,25)(6,13)(7,19,22,21)(10,28,32,24)(12,33)(14,20,26,35)(16,17)(29,31);(5,9,28,33,25,18,32,29,11,30,24,12,8,23,10,31)(6,14,27,19,16,35,15,22,13,26,36,21,17,20,34,7)

Polycyclic presentation:

2^10|a^2=e,b^2=hi,c^2=hi,d^2=efgij,e^2=ij,f^2=j,g^2=j,h^2=j,i^2=j,b^a=befj,c^a=cij,d^a=dfgj,d^b=defgij,d^c=dij,e^b=eh,e^c=ej,e^d=ehij,f^a=fhij,f^b=fhi,f^d=fhij,f^e=fj,g^b=gj,h^a=hj,h^b=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250381
maximal normal subgroup of	#72250381

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