#72306314

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#12852
commutator	C₂²×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂²×#41 ◁ C₂×#3500 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=i,c^2=egj,e^2=i,f^2=j,g^2=j,h^2=j,c^a=cf,c^b=chj,d^a=dfghj,d^c=dfgh,e^c=eh,e^d=ej,f^a=fh,f^b=fij,f^c=fhij,f^e=fij,g^a=ghij,g^b=gij,g^c=gh,g^e=gij,g^f=gj,h^a=hi,h^c=hi,h^f=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306331, #72306355, #72317996, …
maximal normal subgroup of	#72306331, #72306355, #72317996, …

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