#72340596

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂³×C₄
Frattini, cosocle	#390	C₂⁴
prime radical	C₂×#23406	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂×#2297 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|c^2=efhij,e^2=h,f^2=hij,g^2=j,c^a=ce,c^b=cj,d^c=dhi,e^a=eh,e^c=eg,e^d=ej,f^a=fg,f^c=fg,f^e=fij,g^a=gj,g^c=gi,g^e=gj,i^a=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340597, #72340628, #72340736
maximal normal subgroup of	#72340597, #72340628, #72340736

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