#72306391

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#493370
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	#10575549	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#3449 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4539 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)(3,8)(4,18,22,33)(5,10)(6,12)(7,11)(9,21,34,23)(13,27)(14,40,19,49)(15,29)(16,30)(17,39)(20,35,51,48)(24,53,52,32)(25,37)(26,36)(28,42,57,43)(31,47)(38,55)(41,58,50,62)(44,64,54,56)(45,60)(46,61)(59,63);(1,3)(2,8)(4,19)(5,15)(6,17)(7,16)(9,28)(10,29)(11,30)(12,39)(13,31)(14,22)(18,40)(20,50)(21,43)(23,42)(24,54)(25,45)(26,46)(27,47)(32,56)(33,49)(34,57)(35,58)(36,61)(37,60)(38,59)(41,51)(44,52)(48,62)(53,64)(55,63);(1,4)(2,9,12,33)(3,14,16,19)(5,24)(6,23,27,21)(7,22)(8,28,31,40)(10,35,36,53)(11,34,13,18)(15,44,45,54)(17,43)(20,26,51,55)(25,52)(29,58,59,56)(30,57,39,49)(32,37,48,38)(41,63)(42,47)(46,50)(60,62,61,64);(1,5,7,25)(2,10,11,37)(3,15,16,45)(4,20)(6,26,27,55)(8,29,30,60)(9,32)(12,36,13,38)(14,41)(17,46,47,63)(18,48)(19,50)(21,52)(22,51)(23,24)(28,56)(31,59,39,61)(33,35)(34,53)(40,62)(42,54)(43,44)(49,58)(57,64)

Polycyclic presentation:

2^10|a^2=ij,c^2=f,d^2=i,e^2=ij,h^2=j,b^a=bij,c^a=ce,c^b=cg,d^a=dij,d^c=dghij,e^a=ei,e^c=eh,e^d=ej,f^a=fhij,f^b=fj,f^d=fij,f^e=fij,g^a=gj,g^c=gj,h^a=hj,h^c=hi,h^f=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306394, #72306440, #72306877, …
maximal normal subgroup of	#72306394, #72306440, #72306877, …

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