#72322547

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fj,d^2=fghj,e^2=i,f^2=j,g^2=i,h^2=ij,i^2=j,b^a=bh,c^a=cghij,d^a=dfgh,d^b=dgh,d^c=dgh,e^b=ei,e^c=ej,e^d=ej,f^a=fj,g^b=gi,g^c=gj,h^b=hij,h^c=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322570, #72322804
maximal normal subgroup of	#72322570, #72322804

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