#72303126

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#417508
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4842 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)(5,20)(6,12)(7,11)(10,22)(13,26)(14,27)(15,37)(16,28)(17,34)(18,32)(19,31)(21,47)(23,48)(24,29)(25,36)(30,43)(33,49)(35,50)(38,51)(39,59)(40,52)(41,55)(42,60)(44,61)(45,53)(46,58)(54,62)(56,63)(57,64);(1,3)(2,8)(4,13)(5,21)(6,24,25,15)(7,16)(9,26)(10,33)(11,28)(12,37,36,29)(14,23,41,50)(17,42)(18,45,46,39)(19,40)(20,47)(22,49)(27,35,55,48)(30,56)(31,52)(32,59,58,53)(34,60)(38,44,62,64)(43,63)(51,57,54,61);(1,4,7,19)(2,9,11,31)(3,13,16,40)(5,17,22,43)(6,18,25,46)(8,26,28,52)(10,34,20,30)(12,32,36,58)(14,38,41,62)(15,39,24,45)(21,42,49,63)(23,44,50,64)(27,54,55,51)(29,53,37,59)(33,60,47,56)(35,61,48,57);(1,5)(2,10,11,20)(3,14)(4,17)(6,23)(7,22)(8,27,28,55)(9,30,31,34)(12,35,36,48)(13,38)(15,21)(16,41)(18,44)(19,43)(24,49)(25,50)(26,51,52,54)(29,33,37,47)(32,57,58,61)(39,42)(40,62)(45,63)(46,64)(53,56,59,60)

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=e,b^a=bh,c^a=cf,d^a=dfi,d^b=dg,d^c=de,e^a=ej,f^d=fj,g^a=gi,g^b=gi,h^d=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303157, #72303324, #72303440
maximal normal subgroup of	#72303157, #72303324, #72303440

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