#72318261

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 ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5176 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)(3,6)(4,5)(7,8)(12,23)(14,25)(16,30)(18,26);(3,7)(6,8)(9,10)(11,19)(12,16)(13,17)(14,26)(15,22)(18,25)(20,35)(21,34)(23,30)(24,32)(27,31)(28,36)(29,38)(33,37)(39,40);(9,11)(10,15)(12,20,23,37)(13,21)(14,22,25,38)(16,28,30,40)(17,29)(18,19,26,34)(24,36)(27,39)(31,35)(32,33);(2,5)(6,8)(9,12)(10,16)(11,20)(13,23)(14,27)(15,28)(17,30)(18,32)(19,33)(21,37)(22,39)(24,25)(26,31)(29,40)(34,35)(36,38);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(12,23)(18,26)(22,38)(24,27)(28,40)(33,35)

Polycyclic presentation:

2^10|c^2=h,e^2=fi,g^2=j,c^a=ch,c^b=cg,d^a=df,d^b=dij,d^c=di,e^b=efij,e^c=ej,e^d=ej,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

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318342, #72319361
maximal normal subgroup of	#72318342, #72319361

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, 5, F.6*F.9);;
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, 2, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*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);