#72323092

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#357
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₂²×C₁₆	C₂⁴
prime radical	C₂×#33607	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ C₂²×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^10|a^2=g,f^2=hij,g^2=j,h^2=ij,i^2=j,c^b=cj,d^a=df,d^c=de,f^a=fh,f^d=fhj,g^d=gj,h^a=hi,h^d=hi,i^a=ij,i^d=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323143
maximal normal subgroup of	#72323143
maximal quotient of	#72323093

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