#72307336

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1514
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂²×M₁₆	C₂⁴
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=h,c^2=ef,e^2=i,i^2=j,b^a=bj,c^a=ce,c^b=cghi,d^c=dghj,e^a=ei,e^b=ej,e^c=eh,f^a=fh,f^c=fh,f^e=fj,g^c=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307340
maximal normal subgroup of	#72307340

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