#72307226

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

2534 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=gh,e^2=hj,f^2=i,g^2=j,h^2=j,b^a=bh,c^a=cf,d^a=di,d^c=dgh,e^a=ehij,e^b=ej,e^c=ei,f^a=fi,f^c=fij,f^d=fj,h^a=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307250, #72307737
maximal normal subgroup of	#72307250, #72307737

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);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 5, F.8*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);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*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);