#72341086

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7,5,14)(4,11)(6,9)(8,13)(10,16,12,15)(17,18,22,26)(19,23,29,36)(20,24)(21,25)(27,37)(28,43)(30,45,32,38)(31,39)(33,40)(34,42)(35,44)(41,48,46,47);(1,3)(2,7,9,14)(4,12,13,10)(5,6)(8,15)(11,16)(17,19)(18,23)(20,27,31,42)(21,28)(22,29)(24,37,39,34)(25,35)(26,36)(30,41,32,46)(33,44)(38,48,45,47)(40,43);(1,3,6,5)(2,7)(4,10)(8,16,11,15)(9,14)(12,13)(17,20,22,31)(18,24)(19,27)(21,32)(23,34,36,37)(25,38,40,45)(26,39)(28,41,44,46)(29,42)(30,33)(35,48)(43,47);(1,4,6,13)(2,8,9,11)(3,10)(5,12)(7,15)(14,16)(17,21,22,33)(18,25)(19,28)(20,30,31,32)(23,35,36,43)(24,38)(26,40)(27,41)(29,44)(34,47,37,48)(39,45)(42,46);(1,5,6,3)(2,7,9,14)(4,12,13,10)(8,16,11,15)(20,31)(34,37)(38,45)(41,46)

Polycyclic presentation:

2^10|a^2=f,b^2=gj,c^2=hi,d^2=fghi,e^2=i,b^a=bfg,c^a=cfh,c^b=cghi,d^a=dghi,d^b=dfgij,d^c=dfh,e^a=egj,e^b=egij,e^c=eh,e^d=egij,f^b=fi,f^c=fij,f^d=fj,f^e=fi,g^a=gi,g^c=gj,g^d=gi,h^a=hij,h^b=hj,h^d=hij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341091, #72341110, #72341145
maximal normal subgroup of	#72341091, #72341110, #72341145

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