#72303133

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#417821
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5264 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=fj,d^2=ei,b^a=bh,c^a=cf,d^a=de,d^b=dg,d^c=defi,e^b=ej,e^c=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, #72303348, #72303421
maximal normal subgroup of	#72303157, #72303348, #72303421

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