#72342777

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9495
commutator	C₂⁵	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#106542
prime radical	#62811	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

3066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=g,d^2=j,b^a=bf,c^a=ce,c^b=cij,e^b=ei,f^b=fh,f^c=fi,g^a=gh,g^c=gj,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342789, #72342829, #72342843
maximal normal subgroup of	#72342789, #72342829, #72342843

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