#72343769

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

(5,6,10,17,11,18,34,50,12,15,35,46,9,16,31,47)(7,19,23,13,24,39,32,40,21,48,57,37,22,52,58,38)(8,25,29,60,30,14,41,44,27,45,33,64,28,43,63,49)(20,53,55,66,56,51,68,65,54,67,59,42,26,61,62,36);(1,2)(3,4)(5,7)(6,13,15,37)(8,26)(9,24)(10,32)(11,22)(12,21)(14,42,43,66)(16,40,18,38)(17,48,46,19)(20,28)(23,34)(25,36,45,65)(27,56)(29,55)(30,54)(31,57)(33,59)(35,58)(39,47,52,50)(41,62)(44,53,49,67)(51,64,61,60)(63,68);(2,4)(5,8,11,30,12,27,9,28)(6,14,16,25,15,43,18,45)(7,20,24,56,21,54,22,26)(10,33,34,63,35,29,31,41)(13,36,38,42,37,65,40,66)(17,49,47,64,46,44,50,60)(19,51,52,53,48,61,39,67)(23,59,32,62,57,55,58,68);(7,21)(13,37)(19,48)(20,54)(22,24)(23,57)(26,56)(32,58)(36,65)(38,40)(39,52)(42,66)(51,61)(53,67)(55,59)(62,68);(6,15)(8,27)(13,37)(16,18)(17,46)(19,48)(20,54)(26,56)(28,30)(29,33)(38,40)(39,52)(41,63)(47,50)(55,59)(62,68)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=fi,g^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cij,c^b=cg,d^b=dj,e^a=ej,e^c=ej,f^a=fj,g^b=gj,h^b=hi,h^c=hj,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343773, #72343863, #72343873
maximal normal subgroup of	#72343773, #72343863, #72343873

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