#72250831

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21452
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2539 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ef,c^2=ghij,d^2=ghij,e^2=i,h^2=j,i^2=j,b^a=bg,c^a=cehj,c^b=cehj,d^a=dij,d^b=dij,e^b=ej,e^c=ei,e^d=ej,f^b=fj,g^a=gj,h^b=hj,h^c=hj,i^c=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250849, #72252732, #72252746, …
maximal normal subgroup of	#72250849, #72252732, #72252746, …

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