#72224949

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21526
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂	#210728
prime radical	#33617	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #33326 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

943 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=eghi,c^2=hi,d^2=gi,e^2=h,g^2=i,h^2=j,i^2=j,b^a=bg,c^a=cf,d^a=dgi,d^b=dij,e^b=ej,e^d=ej,f^c=fj,g^a=gi,g^b=gj,i^a=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224968, #72224969, #72225088, …
maximal normal subgroup of	#72224968, #72224969, #72225088, …
maximal quotient of	C₂×#72224949

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