#72343485

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂×#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4111 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fhi,d^2=j,f^2=i,h^2=ij,i^2=j,b^a=bi,d^a=dj,d^b=dj,d^c=dg,e^a=ej,e^c=ej,e^d=ej,f^b=fj,h^a=hj,h^b=hj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343493, #72343869
maximal normal subgroup of	#72343493, #72343869

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