#72321228

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	C₂×#3414	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=hj,f^2=h,g^2=i,i^2=j,b^a=bgi,c^b=cj,d^a=dj,d^b=dh,e^a=eg,e^b=eh,e^c=ej,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321406, #72321717
maximal normal subgroup of	#72321406, #72321717

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