#72343401

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×#525	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4127 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fhi,f^2=i,h^2=ij,i^2=j,b^a=bi,d^a=dj,d^b=dj,d^c=dg,e^c=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	#72343407, #72343421, #72343869
maximal normal subgroup of	#72343407, #72343421, #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, 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, 3, 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);