#72293126

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2127
commutator	C₄²	C₂²×C₄²
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	C₂×#3151	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #34070 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|c^2=e,d^2=g,e^2=j,f^2=ij,h^2=j,c^a=cf,c^b=ci,d^a=di,e^a=eh,e^b=ej,f^a=fij,f^c=fhij,f^d=fj,f^e=fi,h^a=hj,h^c=hi,h^e=hj,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293129, #72293217
maximal normal subgroup of	#72293129, #72293217

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