#72303120

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10960
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×D₄	C₂⁴
socle	C₂	C₂×#10960
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#3428 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5533 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)(3,13,16,35)(4,18)(5,10,22,30)(6,12)(7,11)(8,17,27,42)(9,21)(14,36,38,33)(15,34)(19,44,46,28)(20,45)(23,31,49,55)(24,32)(25,50,43,39)(26,52)(29,48)(37,58)(40,57)(41,53)(47,62,63,54)(51,60)(56,59)(61,64);(1,3)(2,8)(4,14)(5,15)(6,17,24,42)(7,16)(9,25)(10,26)(11,27)(12,13,32,35)(18,43)(19,37)(20,39,48,50)(21,38)(22,40)(23,41,49,52)(28,51)(29,33,45,36)(30,53)(31,34,55,57)(44,61)(46,59)(47,60,63,64)(54,56,62,58);(1,4,7,21)(2,9,11,18)(3,14,16,38)(5,19,22,46)(6,20,24,48)(8,25,27,43)(10,28,30,44)(12,29,32,45)(13,33,35,36)(15,37,40,59)(17,39,42,50)(23,47,49,63)(26,51,53,61)(31,54,55,62)(34,56,57,58)(41,60,52,64);(1,5,7,22)(2,10)(3,15,16,40)(4,19,21,46)(6,23,24,49)(8,26)(9,28)(11,30)(12,31)(13,34)(14,37,38,59)(17,41,42,52)(18,44)(20,47,48,63)(25,51)(27,53)(29,54)(32,55)(33,56)(35,57)(36,58)(39,60,50,64)(43,61)(45,62)

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=j,d^2=fi,g^2=j,b^a=bg,c^a=cj,d^a=def,e^b=ej,e^d=ej,f^a=fj,f^b=fj,g^b=gi,h^a=hij,h^g=hj,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303123, #72303302, #72303421
maximal normal subgroup of	#72303123, #72303302, #72303421

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