#72321860

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21452
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2173 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)(5,6,12,20,10,25,11,19,14,21,15,24,13,18,34,7)(8,26,28,31,17,33,27,36,30,16,35,32,29,9,23,22);(1,2)(3,4)(5,7,10,24,14,19,13,20)(6,15,18,34,21,12,25,11)(8,22,17,32,30,36,29,31)(9,23,16,28,33,27,26,35);(2,4)(5,8,13,29,14,30,10,17)(6,16,25,9,21,26,18,33)(7,22,20,31,19,36,24,32)(11,23,12,35,34,27,15,28);(2,4)(5,9,10,16,14,33,13,26)(6,17,18,30,21,29,25,8)(7,23,24,28,19,27,20,35)(11,22,15,32,34,36,12,31)

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=ej,d^2=e,e^2=j,f^2=i,g^2=ij,h^2=i,i^2=j,b^a=bf,c^a=ceghi,c^b=cfgj,d^a=dehj,d^b=dfgj,d^c=dj,e^a=ej,f^b=fi,f^c=fj,f^d=fi,g^b=gij,g^c=gj,g^d=gij,h^b=hi,h^d=hij,i^b=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321926, #72322333
maximal normal subgroup of	#72321926, #72322333

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