#72285609

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁵
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4905 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:

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

Polycyclic presentation:

2^10|a^2=ij,c^2=g,f^2=hj,g^2=i,h^2=j,c^a=cg,c^b=cf,d^a=di,d^b=dij,d^c=dij,e^a=ei,e^b=eij,f^a=fi,f^b=fh,f^c=fhi,g^a=gi,g^b=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285790
maximal normal subgroup of	#72285790

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