#72286355

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂³×D₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₂×C₄²×D₄
prime radical	C₂×#575	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4305 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=f,e^2=fgij,f^2=i,g^2=ij,h^2=j,b^a=bh,c^a=ch,c^b=chj,d^a=dj,d^b=dj,e^b=ej,e^c=ei,h^a=hj,h^c=hj

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286550, #72322246
maximal normal subgroup of	#72286550, #72322246

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