#72264419

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#410
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#117	C₂⁴
socle	C₂²	C₂×#1799
prime radical	C₂×#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

2150 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:

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

Polycyclic presentation:

2^10|b^2=ef,c^2=g,e^2=i,g^2=j,b^a=be,c^a=ci,c^b=cg,d^b=dij,e^a=ei,e^b=eh,f^a=fh,f^b=fh,f^e=fj,g^b=gj,h^b=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264428, #72264470, #72264478, …
maximal normal subgroup of	#72264428, #72264470, #72264478, …

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