#72307178

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	C₄	#21452
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#354257
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2769 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fgij,c^2=efhij,d^2=efgj,e^2=ij,f^2=i,g^2=i,h^2=j,i^2=j,b^a=bf,c^a=ci,d^a=dhij,d^b=dfhij,d^c=dij,e^b=ei,e^c=ej,f^a=fj,f^b=fi,f^c=fj,g^b=gi,g^c=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72319811, #72320350
maximal normal subgroup of	#72307301, #72319811, #72320350

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