#72283967

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
prime radical	#245228	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2748 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,18,23,19)(20,24,32,27)(21,28,35,25)(22,26,37,29)(30,40,33,38)(31,39,46,43)(34,41,36,42)(44,47,45,48);(1,3,6,11)(2,7,5,9)(4,12,13,10)(8,16,14,15)(17,19,23,18)(20,27,32,24)(21,25,35,28)(22,26,37,29)(30,38,33,40)(31,43,46,39)(34,42,36,41)(44,47,45,48);(1,4)(2,8)(3,10,11,12)(5,14)(6,13)(7,15,9,16)(17,20)(18,24)(19,27)(21,33,35,30)(22,31)(23,32)(25,40,28,38)(26,39)(29,43)(34,45,36,44)(37,46)(41,48,42,47);(1,5,6,2)(3,9,11,7)(4,8)(10,15)(12,16)(13,14)(17,21)(18,25)(19,28)(20,30)(22,36,37,34)(23,35)(24,38)(26,42,29,41)(27,40)(31,45,46,44)(32,33)(39,48,43,47);(7,9)(15,16)(17,22)(18,26)(19,29)(20,31)(21,34)(23,37)(24,39)(25,41)(27,43)(28,42)(30,44)(32,46)(33,45)(35,36)(38,47)(40,48)

Polycyclic presentation:

2^10|a^2=i,b^2=ij,c^2=f,d^2=hj,b^a=bj,c^b=ch,d^a=dhij,d^b=di,d^c=dg,e^a=ef,e^b=egi,e^d=ef,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72285772, #72340254, …
maximal normal subgroup of	#72284709, #72285772, #72340254, …

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