#72343149

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2847
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	#14697
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

2051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=eij,c^2=ei,e^2=i,h^2=j,b^a=bh,c^a=cg,c^b=cgj,d^a=df,d^b=df,d^c=di,f^c=fi,g^d=gi,h^a=hj,h^b=hj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343153, #72343160, #72343234
maximal normal subgroup of	#72343153, #72343160, #72343234

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