#72319286

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

2568 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)(11,17)(12,19)(13,20)(14,21)(15,18)(16,22)(23,32)(24,40)(25,34)(26,35)(27,36)(28,38)(29,37)(30,33)(31,39);(2,7)(4,8)(11,23)(17,32)(24,30)(25,27)(26,31)(33,40)(34,36)(35,39);(1,2,5,8,6,7,3,4)(9,11,13,26,15,23,16,31)(10,17,20,35,18,32,22,39)(12,25,28,24,29,27,14,30)(19,34,38,40,37,36,21,33);(1,3,6,5)(2,8,7,4)(10,18)(11,24)(12,14)(13,16)(17,33)(19,38)(21,37)(23,30)(25,26)(27,31)(28,29)(32,40)(34,39)(35,36);(1,4,3,7,6,8,5,2)(9,12,16,14,15,29,13,28)(10,19,22,21,18,37,20,38)(11,25,31,30,23,27,26,24)(17,34,39,33,32,36,35,40)

Polycyclic presentation:

2^10|c^2=f,d^2=hj,e^2=fh,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dj,d^c=dg,e^b=ehj,e^d=ehij,f^d=fj,g^c=gi,g^d=gij,g^e=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319390, #72319510, #72319699, …
maximal normal subgroup of	#72319390, #72319510, #72319699, …

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