#72281205

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346542
commutator, Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10583463	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3718 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)(3,10)(4,16)(5,25)(6,31)(7,35)(8,15)(9,11)(12,27)(13,20)(14,32)(17,42)(18,52)(19,53)(21,41)(22,38)(23,50)(24,51)(26,47)(28,62)(29,60)(30,61)(33,43)(34,40)(36,46)(37,44)(39,55)(45,58)(48,56)(49,59)(54,63)(57,64);(1,3)(2,10)(4,22)(5,26)(6,32)(7,24)(8,21)(9,17)(11,42)(12,45)(13,44)(14,31)(15,41)(16,38)(18,29)(19,34)(20,37)(23,57)(25,47)(27,58)(28,59)(30,54)(33,36)(35,51)(39,48)(40,53)(43,46)(49,62)(50,64)(52,60)(55,56)(61,63);(1,4,8,9)(2,11,15,16)(3,17,21,22)(5,18,30,57)(6,20,34,24)(7,19,37,32)(10,38,41,42)(12,46,49,39)(13,31,51,40)(14,44,53,35)(23,26,29,54)(25,52,61,64)(27,36,59,55)(28,56,58,33)(43,62,48,45)(47,60,63,50);(1,5,4,23,8,30,9,29)(2,12,11,43,15,49,16,48)(3,18,22,26,21,57,17,54)(6,33,24,27,34,56,20,59)(7,36,19,58,37,55,32,28)(10,39,42,45,41,46,38,62)(13,50,40,25,51,60,31,61)(14,52,44,63,53,64,35,47)

Polycyclic presentation:

2^10|c^2=j,d^2=hi,e^2=ij,f^2=i,g^2=j,i^2=j,c^a=ch,c^b=ch,d^a=de,d^b=dg,d^c=defij,e^a=ei,e^b=ej,e^c=ej,e^d=eij,f^a=fi,f^b=fh,f^d=fi,g^a=gj,g^b=gj,g^c=ghj,g^d=gj,g^f=gj,h^d=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281944, #72307300
maximal normal subgroup of	#72281944, #72307300

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