#72280974

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#109648
commutator	C₂²×C₈	C₂×C₄²
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

1111 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)(3,13,15,37)(4,18)(5,24,26,34)(6,12)(7,27)(8,16)(9,23)(10,25,35,52)(11,20)(14,36,40,61)(17,47,41,39)(19,33)(21,32,53,60)(22,31)(28,45)(29,44)(30,58)(38,48,62,43)(42,49)(46,56)(50,54,55,51)(57,64)(59,63);(3,14,15,40)(4,19)(5,25)(7,20)(8,28,29,57)(9,31)(10,34)(11,27)(13,36,37,61)(16,45,44,64)(17,48,41,43)(18,33)(22,23)(24,35)(26,52)(30,59,56,49)(38,47,62,39)(42,58,63,46)(50,55)(51,54);(1,3,4,14,20,41,19,48,6,15,22,40,7,17,23,43)(2,8,9,28,27,56,31,59,12,29,33,57,11,30,18,49)(5,16,21,42,52,58,51,64,26,44,53,63,25,46,54,45)(10,13,32,38,24,47,55,61,35,37,60,62,34,39,50,36)

Polycyclic presentation:

2^10|a^2=d,b^2=dg,c^2=e,e^2=ij,f^2=i,i^2=j,c^a=cf,c^b=ce,d^c=dh,e^a=ei,e^b=eij,f^a=fhij,f^b=fi,f^d=fj,g^c=ghj,g^f=gj,h^a=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281197, #72281396, #72282273
maximal normal subgroup of	#72281197, #72281396, #72282273

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