#72318010

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄² ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|c^2=h,g^2=j,c^a=ch,c^b=cg,d^a=df,d^b=dij,d^c=di,e^a=efij,e^b=efi,e^c=ej,e^d=efij,f^c=fj,g^a=gi,g^b=gj,g^c=gi,g^d=gj,h^b=hij,h^d=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318094, #72319015, #72319433
maximal normal subgroup of	#72318094, #72319015, #72319433

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