#72307907

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#24061
prime radical	#10575347	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,c^2=fghj,f^2=ij,h^2=j,i^2=j,b^a=bg,c^a=cf,c^b=cfhj,d^a=dg,d^b=dg,d^c=dg,e^d=ej,f^a=fi,f^b=fij,f^c=fj,f^d=fj,g^c=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308106
maximal normal subgroup of	#72308106

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