#72311591

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

5899 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)(3,6,8,4)(10,15)(11,19)(16,28)(21,34)(22,35)(23,32)(36,39)(38,40);(1,3,5,8)(2,6,7,4)(9,10)(11,20)(12,23)(13,26,27,18)(14,15)(16,22,28,35)(17,31,29,25)(19,33)(21,37)(24,32)(30,34)(36,40,39,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,21)(13,22)(14,19)(15,28)(17,30)(18,20)(23,39)(24,34)(25,38)(26,33)(27,35)(29,37)(31,40)(32,36);(1,4,5,6)(2,3,7,8)(10,15)(11,19)(12,24)(18,26)(22,35)(25,31)(30,37)(36,39);(2,7)(3,8)(9,12,14,24)(10,17,15,29)(11,21,19,34)(13,25,27,31)(16,30,28,37)(18,32,26,23)(20,36,33,39)(22,38,35,40)

Polycyclic presentation:

2^10|a^2=f,b^2=fhj,d^2=f,e^2=j,b^a=bh,c^a=cij,c^b=cfg,d^b=dfj,d^c=dj,e^a=efi,e^b=efgj,e^c=ef,e^d=ej,g^a=gi,g^b=gi,h^c=hi,h^e=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311623, #72312100
maximal normal subgroup of	#72311623, #72312100

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