#72319797

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#356085
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#576	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ C₂×#1757 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3626 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)(3,15,18,43)(4,21,20,51)(5,25,27,11)(6,30)(7,33)(8,13)(9,19,37,31)(10,24,39,46)(12,32)(14,34)(16,44)(17,48)(22,35)(23,54)(26,52)(28,61,53,64)(29,42,55,59)(36,49)(38,56)(40,60)(41,63,58,47)(45,50)(57,62);(1,3,16,46,34,4,22,19,8,18,32,24,7,20,6,31)(2,9,35,21,33,39,44,15,13,37,30,51,14,10,12,43)(5,17,47,60,55,23,53,49,27,50,63,56,29,26,28,62)(11,36,61,52,59,40,58,45,25,57,64,54,42,38,41,48);(1,4,7,18,8,20,34,3)(2,10,14,9,13,39,33,37)(5,26,29,17,27,23,55,50)(6,19,32,24,22,31,16,46)(11,40,42,57,25,38,59,36)(12,43,30,21,44,15,35,51)(28,62,63,60,53,49,47,56)(41,45,64,54,58,48,61,52);(1,5,8,27)(2,11,13,25)(3,17,18,50)(4,23,20,26)(6,28,22,53)(7,29,34,55)(9,36,37,57)(10,38,39,40)(12,41,44,58)(14,42,33,59)(15,45,43,48)(16,47,32,63)(19,49,31,62)(21,52,51,54)(24,56,46,60)(30,64,35,61)

Polycyclic presentation:

2^10|a^2=e,b^2=efhij,c^2=ehi,d^2=j,f^2=i,i^2=j,b^a=bf,c^a=cf,c^b=cfj,d^a=dg,d^c=dj,e^b=ej,e^c=ej,e^d=ej,f^a=fi,f^b=fj,f^c=fi,g^a=gj,h^d=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319811, #72321054, #72343980
maximal normal subgroup of	#72319811, #72321054, #72343980

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