#72340499

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2773 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)(4,8)(5,11)(6,10)(9,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,32)(22,24)(26,33)(27,35)(28,39)(29,34)(30,38)(31,37)(36,42)(40,45)(41,46)(43,44)(47,48);(1,3,6,12)(2,7,10,14)(4,9)(5,8)(11,13)(15,16)(17,19)(18,23)(20,27)(21,28,33,42)(22,29)(24,34)(25,35)(26,39,32,36)(30,41,43,48)(31,40)(37,45)(38,47,44,46);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(18,24)(21,33)(23,34)(26,32)(27,40)(35,45)(38,44)(46,47);(1,5,6,15)(2,9)(3,8,12,16)(4,7)(10,11)(13,14)(17,20)(18,25,24,37)(19,27,29,40)(21,30)(22,31)(23,35)(26,38,32,44)(28,41,42,48)(33,43)(34,45)(36,46)(39,47);(2,10)(3,12)(5,15)(7,14)(17,21)(18,26)(19,28)(20,30)(22,33)(23,36)(24,32)(25,38)(27,41)(29,42)(31,43)(34,39)(35,46)(37,44)(40,48)(45,47)

Polycyclic presentation:

2^10|b^2=fi,d^2=gi,b^a=bf,c^a=cfh,c^b=cfhj,d^a=dfh,d^b=dhi,d^c=dfg,e^a=ehi,e^b=egij,e^c=egh,e^d=ei,f^c=fi,f^d=fi,f^e=fj,g^a=gij,g^b=gj,g^c=gi,h^b=hi,h^c=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340502
maximal normal subgroup of	#72340502

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