#72304154

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×B(4,𝔽₂)
commutator	C₂×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×#117	C₂⁴
socle	C₂²	C₂×#1790
prime radical	C₂×#36652	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

5494 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:

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

Polycyclic presentation:

2^10|b^2=j,c^2=e,e^2=j,f^2=i,b^a=bhj,c^a=cg,c^b=cf,d^c=di,e^b=eh,f^a=fh,f^b=fi,f^c=fhi,f^e=fj,g^b=gh,g^f=gj,h^c=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304155
maximal normal subgroup of	#72304155

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