#72280909

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#17595
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂⁴
socle	C₂	#182644
prime radical	#60450	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2021 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)(3,14)(4,20)(5,26)(6,28,30,58)(7,11)(8,13)(9,17,35,49)(10,23)(12,31,40,27)(15,39)(16,46)(18,42)(19,43)(21,41)(22,50)(24,55,53,38)(25,37)(29,32,45,34)(33,52,60,64)(36,47,59,56)(44,61)(48,63)(51,57,62,54);(1,3,5,15,8,19,7,18)(2,9,11,34,13,35,26,32)(4,21,23,16,22,44,25,48)(6,29,27,17,30,45,31,49)(10,36,20,33,37,59,50,60)(12,39,28,14,40,42,58,43)(24,56,54,52,53,47,57,64)(38,61,51,46,55,41,62,63);(3,15,19,18)(4,22)(9,32,35,34)(10,37)(14,39,43,42)(16,21,48,44)(17,45,49,29)(20,50)(23,25)(24,53)(33,59,60,36)(38,55)(41,63,61,46)(47,52,56,64)(51,62)(54,57);(1,4,5,23,8,22,7,25)(2,10,11,20,13,37,26,50)(3,16,18,21,19,48,15,44)(6,24,27,54,30,53,31,57)(9,33,32,36,35,60,34,59)(12,38,28,51,40,55,58,62)(14,41,39,46,43,61,42,63)(17,47,29,52,49,56,45,64)

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=eij,d^2=i,e^2=j,f^2=i,h^2=j,i^2=j,b^a=bf,c^b=chj,d^a=di,d^b=di,d^c=dj,e^a=ej,f^a=fh,f^b=fi,g^b=ghi,g^f=gj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281197, #72281382, #72281673
maximal normal subgroup of	#72281197, #72281382, #72281673

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