#72318102

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	C₂×#12759
commutator	#127	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	#10576177	C₂
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5912 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)(4,9)(5,22,25,10)(6,27)(7,11)(12,29)(13,34)(14,44,46,38)(15,48)(16,32)(17,33)(18,30)(19,35,24,39)(20,36)(21,54)(23,31,47,60)(26,41,28,40)(37,53)(42,64,45,61)(43,57)(49,58)(50,56,55,59)(51,62,52,63);(1,3)(2,8)(4,18,20,13)(5,23,26,46)(6,16,29,15)(7,17)(9,34,36,30)(10,38,40,31)(11,33)(12,32,27,48)(14,25,47,28)(19,50,51,42)(21,49)(22,44,41,60)(24,55,52,45)(35,61,62,59)(37,57)(39,64,63,56)(43,53)(54,58);(1,4)(2,9)(3,13)(5,24)(6,21)(7,20)(8,30)(10,39)(11,36)(12,37)(14,45)(15,49)(16,43)(17,18)(19,25)(22,35)(23,55)(26,51)(27,54)(28,52)(29,53)(31,59)(32,58)(33,34)(38,64)(40,62)(41,63)(42,46)(44,61)(47,50)(48,57)(56,60);(1,5,7,25)(2,10,11,22)(3,14,17,46)(4,19,20,24)(6,28,29,26)(8,31,33,60)(9,35,36,39)(12,41,27,40)(13,42,18,45)(15,47,16,23)(21,51,53,52)(30,56,34,59)(32,44,48,38)(37,62,54,63)(43,50,49,55)(57,61,58,64)

Polycyclic presentation:

2^10|a^2=e,b^2=g,d^2=j,h^2=j,b^a=beg,c^a=cij,c^b=cij,d^a=def,d^b=defj,d^c=dj,e^d=ej,f^a=fj,f^b=fh,f^c=fj,f^d=fj,g^c=gj,g^d=ghj,g^f=gij,h^a=hi,h^b=hi,h^d=hj,h^f=hj,h^g=hj,i^b=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318143, #72318336, #72318618
maximal normal subgroup of	#72318143, #72318336, #72318618

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