#72306865

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#128033
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂×#536	C₂³
prime radical	C₂×#3499	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ C₂×#3499 ◁ G
LECS	G ▷ C₂×#536 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2167 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=ij,c^2=e,d^2=ij,e^2=j,f^2=j,g^2=j,h^2=j,b^a=bj,c^a=cf,c^b=cgi,d^c=dg,e^a=eghj,e^d=ej,f^a=fg,f^b=fij,f^c=fgh,f^d=fij,f^e=fj,g^a=gi,g^f=gj,h^a=hij,h^c=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306881
maximal normal subgroup of	#72306881

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