#72280028

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2297
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	#13956
prime radical	#7629700	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

4911 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)(9,10)(11,16)(12,22)(13,17,27,35)(14,19)(15,30)(18,29)(20,31,23,39)(21,38)(24,32)(25,34,26,33)(28,40,37,36);(2,6)(3,4)(7,8)(9,11,14,12)(10,16,19,22)(13,25)(15,21,29,24)(17,33)(18,32,30,38)(20,37)(23,28)(26,27)(31,40)(34,35)(36,39);(9,12,14,11)(10,16,19,22)(13,26)(15,24,29,21)(17,34)(18,32,30,38)(20,28)(23,37)(25,27)(31,36)(33,35)(39,40);(1,2,4,7,5,8,3,6)(9,13)(10,17)(11,20,12,23)(14,27)(15,28)(16,31,22,39)(18,36)(19,35)(21,25,24,26)(29,37)(30,40)(32,33,38,34)

Polycyclic presentation:

2^10|a^2=g,b^2=gij,c^2=gij,d^2=f,e^2=j,f^2=j,h^2=i,b^a=bg,c^a=cij,d^a=de,d^b=dh,d^c=dehi,e^a=ei,e^b=ej,f^a=fj,f^b=fi,f^c=fij,g^d=gij,h^a=hi,h^b=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280131, #72280226
maximal normal subgroup of	#72280131, #72280226

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