#72306439

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	C₂×#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5909 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)(3,6)(4,5)(7,14)(8,11)(9,10)(12,13)(15,16)(17,18,23,27)(19,28)(20,25)(21,26,36,44)(22,37)(24,31)(29,40,32,38)(30,42)(33,41)(34,39)(35,43)(45,48,46,47);(1,3,7,10,8,13,5,15)(2,9,4,6,11,16,14,12)(17,19,23,31)(18,24)(20,32)(21,34,36,30)(22,29)(25,40,37,38)(26,42)(27,28)(33,45)(35,46)(39,44)(41,47,43,48);(1,4)(2,5)(3,12)(6,13)(7,11)(8,14)(9,10)(15,16)(17,20,23,22)(18,25)(19,29)(21,35,36,33)(24,38,28,40)(26,43)(27,37)(30,46)(31,32)(34,45)(39,48,42,47)(41,44);(1,5,8,7)(2,4,11,14)(3,10,13,15)(6,9,12,16)(17,21,23,36)(18,26,27,44)(19,30,31,34)(20,33,22,35)(24,39,28,42)(25,41,37,43)(29,45,32,46)(38,47,40,48);(1,6,7,16,8,12,5,9)(2,10,4,13,11,15,14,3)(17,22,23,20)(18,25,27,37)(19,29,31,32)(21,35,36,33)(24,38,28,40)(26,41,44,43)(30,45,34,46)(39,47,42,48)

Polycyclic presentation:

2^10|a^2=f,b^2=fgi,c^2=fghj,d^2=ij,e^2=fhi,g^2=j,h^2=j,b^a=bg,c^a=cghj,c^b=cfj,d^b=dij,d^c=dij,e^a=egij,e^b=efgh,e^c=efghij,f^b=fi,f^c=fi,f^e=fi,g^a=gij,g^b=gi,h^a=hj,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306523, #72306526
maximal normal subgroup of	#72306523, #72306526

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