#72343045

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2433
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	#14189
prime radical	#235047	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

2508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,8,5,7,3,6)(9,10,13,18,14,19,15,26)(11,16,22,30,23,20,27,34)(12,17,24,31,25,32,28,37)(21,29,35,39,36,33,38,40);(2,6)(3,4)(7,8)(9,10,15,26,14,19,13,18)(11,20,27,30,23,16,22,34)(12,17,28,37,25,32,24,31)(21,29,38,40,36,33,35,39);(2,7)(6,8)(9,11,15,27,14,23,13,22)(10,16,26,34,19,20,18,30)(12,21,24,35,25,36,28,38)(17,29,31,39,32,33,37,40);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,17,19,32)(11,21)(13,24,15,28)(16,29)(18,31,26,37)(20,33)(22,35)(23,36)(27,38)(30,39)(34,40)

Polycyclic presentation:

2^10|a^2=e,b^2=ehj,c^2=eghi,d^2=fij,e^2=i,h^2=j,b^a=bh,c^a=cj,c^b=cg,d^b=df,d^c=dfgi,e^b=ej,f^c=fij,g^d=gij,h^b=hj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343046, #72343167, #72343216
maximal normal subgroup of	#72343046, #72343167, #72343216

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