#72257189

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31681
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1742 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=hj,d^2=fh,e^2=i,f^2=j,i^2=j,b^a=bg,c^a=ceh,d^a=df,d^b=dhj,d^c=dfgi,e^c=ei,e^d=ej,f^c=fj,g^d=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257203, #72257342, #72257690
maximal normal subgroup of	#72257203, #72257342, #72257690

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