#72264458

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#99
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂×#117	C₂⁴
socle	C₂²	C₂×C₄×#99
prime radical	C₂×#392	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

2258 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)(10,16)(18,30)(19,26)(20,32)(21,36)(29,31)(35,40)(37,39);(1,2,3,7,5,8,4,6)(9,10,17,32)(11,21,33,39)(12,26,14,20)(13,29,28,16)(15,19,27,31)(18,34,40,25)(22,30,24,37)(23,35,38,36);(1,3,5,4)(2,7,8,6)(9,11,14,24,15,25,13,23)(10,18,20,35,19,21,29,37)(12,22,27,34,28,38,17,33)(16,30,32,40,26,36,31,39);(2,8)(6,7)(10,19)(16,26)(18,21)(20,29)(30,36)(31,32)(35,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264470
maximal normal subgroup of	#72264470

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