#72321741

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	#34963
prime radical	#411663	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4257 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)(5,6,10,16,12,17,11,26)(7,13,20,30,21,28,22,35)(8,14,23,31,24,15,9,25)(18,27,32,36,33,29,19,34);(1,2)(3,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,22)(12,21)(14,27)(15,29)(16,30)(17,28)(23,32)(24,33)(25,34)(26,35)(31,36);(5,8,11,9,12,24,10,23)(6,14,26,25,17,15,16,31)(7,18,22,19,21,33,20,32)(13,27,35,34,28,29,30,36);(1,3,4,2)(13,28)(27,29)(30,35)(34,36);(5,9,11,24,12,23,10,8)(6,15,16,25,17,14,26,31)(7,19,22,33,21,32,20,18)(13,29,30,34,28,27,35,36)

Polycyclic presentation:

2^10|a^2=f,c^2=fj,d^2=g,e^2=i,f^2=j,i^2=j,b^a=bg,d^a=dh,d^b=dfgi,e^a=ei,e^c=ej,h^b=hj,i^a=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321803, #72322302, #72322646, …
maximal normal subgroup of	#72321803, #72322302, #72322646, …

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