#72226709

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

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	C₂²×C₄×C₂²⋊C₄
prime radical	C₂³×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

8847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=fgi,c^2=h,e^2=j,f^2=j,g^2=i,h^2=j,b^a=bf,d^a=dij,d^b=dij,d^c=dj,e^b=ej,f^a=fj

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226716
maximal normal subgroup of	#72226716

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