#72307229

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	#34963
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2522 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=ghi,c^2=ghi,e^2=hij,f^2=i,g^2=j,h^2=j,b^a=bh,c^a=cf,d^a=di,d^c=dghi,e^a=ehj,e^b=ej,e^c=ei,f^a=fi,f^c=fij,f^d=fj,h^a=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307250, #72307751
maximal normal subgroup of	#72307250, #72307751

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