#72320672

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#392585
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3584	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #60058 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5881 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=ej,f^2=i,i^2=j,b^a=bf,c^a=ch,c^b=ci,d^a=dg,e^b=ej,f^a=fi,f^b=fij,f^c=fj,g^c=gj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320733, #72321198, #72343936
maximal normal subgroup of	#72320733, #72321198, #72343936
maximal quotient of	#72343251

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