#72318508

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×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5861 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)(4,8)(5,7)(13,14)(16,31)(18,24)(21,25)(27,28)(32,37)(34,38)(36,40);(1,2)(3,6)(4,8)(5,7)(9,10,15,17)(11,20,23,33)(12,24)(13,19)(14,29)(16,30)(18,26)(21,35)(22,31)(25,39)(27,38,28,34)(32,36,37,40);(1,3,5,8)(2,6,7,4)(9,11,14,30)(10,16,29,33)(12,25,27,40)(13,22,15,23)(17,31,19,20)(18,32,34,35)(21,28,36,26)(24,37,38,39);(1,4,5,6)(2,3,7,8)(10,17)(16,31)(18,24)(19,29)(20,33)(32,37)(34,38)(35,39);(2,7)(3,8)(9,12)(10,18)(11,21)(13,27)(14,28)(15,26)(16,32)(17,24)(19,34)(20,35)(22,36)(23,25)(29,38)(30,40)(31,37)(33,39)

Polycyclic presentation:

2^10|b^2=fj,c^2=fgj,d^2=ghij,g^2=j,h^2=j,b^a=bf,c^a=cf,c^b=ch,d^a=dghij,d^b=dghi,d^c=dghij,e^a=egh,e^b=egh,e^c=egh,f^c=fi,f^e=fj,g^a=gi,g^b=gi,g^c=gi,g^f=gj,h^a=hi,h^b=hij,h^c=hij,h^f=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318560, #72319313, #72319813, …
maximal normal subgroup of	#72318560, #72319313, #72319813, …

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