#72342797

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	#8541
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#106132
prime radical	#62811	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

3715 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=ef,d^2=j,b^a=be,c^a=cg,c^b=cgj,d^b=di,d^c=dh,e^a=ei,e^b=ei,f^b=fi,f^c=fh,g^a=gh,g^b=gh,g^e=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342803, #72342849, #72342852
maximal normal subgroup of	#72342803, #72342849, #72342852

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