#72321220

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32002
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3067	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1499 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efj,d^2=ghi,f^2=j,g^2=i,h^2=i,i^2=j,b^a=bg,c^a=ch,c^b=cfi,d^a=dhj,d^b=defgh,d^c=dfghj,e^b=ej,e^c=ej,e^d=ej,f^a=fj,f^b=fj,f^c=fj,f^d=fj,g^a=gi,g^b=gi,g^c=gij,g^d=gij,h^a=hi,h^b=hij,h^c=hij,h^d=hij,i^a=ij,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321276, #72322256, #72322302, …
maximal normal subgroup of	#72321276, #72322256, #72322302, …

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