#72321854

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23573
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6630068	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2375 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=ej,d^2=eij,e^2=j,f^2=i,g^2=i,h^2=ij,i^2=j,b^a=bf,c^a=ci,c^b=cfh,d^a=dfgij,d^b=defg,e^a=ej,f^b=fi,f^c=fj,g^b=gij,g^c=gj,g^d=gj,h^b=hij,h^c=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321960, #72322329
maximal normal subgroup of	#72321960, #72322329

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