#72322193

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	#30396
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#33618	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2032 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)(5,6,13,19,26,17,34,24,14,21,12,7,10,20,11,18)(8,9,29,22,27,33,28,32,16,15,35,25,30,31,23,36);(1,2)(3,4)(5,7,10,18,14,19,26,24)(6,11,17,12,21,34,20,13)(8,25,27,32,16,22,30,36)(9,28,31,35,15,23,33,29);(2,4)(5,8,26,27,14,16,10,30)(6,15,17,31,21,9,20,33)(7,22,18,32,19,25,24,36)(11,23,13,29,34,28,12,35);(2,4)(5,9,26,31,14,15,10,33)(6,16,17,27,21,8,20,30)(7,23,18,35,19,28,24,29)(11,22,13,36,34,25,12,32)

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=ij,d^2=ghj,e^2=j,f^2=i,g^2=ij,h^2=ij,i^2=j,b^a=bf,c^a=cj,c^b=cefgij,d^a=dhj,d^b=defhi,d^c=dghij,e^a=ej,e^c=ej,e^d=ej,f^b=fi,f^c=fj,f^d=fi,g^b=gij,g^c=gj,g^d=gij,h^b=hij,h^d=hi,i^b=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322221, #72322333, #72322428, …
maximal normal subgroup of	#72322221, #72322333, #72322428, …

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