#72307662

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#18194
commutator, Frattini, cosocle	C₂×#97	C₂⁴
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=gh,b^2=fghi,c^2=gh,e^2=h,f^2=i,i^2=j,b^a=bf,c^a=cgh,c^b=ce,d^c=dgh,e^a=eg,e^b=egh,e^c=ehj,e^d=ej,f^a=fi,f^b=fj,f^c=fg,f^e=fj,g^b=gj,i^a=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307671, #72307707, #72307788
maximal normal subgroup of	#72307671, #72307707, #72307788

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