#72280087

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	#18186
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂²×M₁₆	C₂⁴
prime radical	C₂×#33692	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1898 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=ghi,c^2=hij,d^2=h,e^2=ij,h^2=j,i^2=j,c^a=cj,c^b=chi,d^a=de,d^b=deij,d^c=def,e^a=ei,e^b=eij,e^c=eh,e^d=eij,f^a=fhi,f^b=fhij,f^c=fhij,f^d=fj,f^e=fj,h^a=hj,h^b=hj,h^c=hj,i^a=ij,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280113
maximal normal subgroup of	#72280113

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