#72280964

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂⁴
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×D₄² ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4302 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:

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

Polycyclic presentation:

2^10|b^2=j,c^2=hj,d^2=eij,e^2=i,f^2=i,g^2=j,b^a=bj,c^a=ch,c^b=chj,d^a=de,d^b=dg,d^c=defi,e^a=ei,e^b=ej,f^a=fij,f^b=fh,f^c=fj,g^a=gj,g^b=gj,g^c=ghj,g^f=gj,h^d=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281530
maximal normal subgroup of	#72281530

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