#72319192

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×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5047 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(20,31)(21,33)(23,36)(24,38)(25,40)(27,41)(37,48)(43,44);(2,7)(8,14)(9,10)(15,16)(17,18)(19,27,29,41)(20,24)(21,25)(22,26)(23,30,36,34)(28,44,45,43)(31,38)(32,39)(33,40)(35,42)(37,46,48,47);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,23,26,36)(20,28,31,45)(21,34,33,30)(22,29)(24,37)(25,41)(27,40)(32,47)(35,46)(38,48)(39,43,42,44);(1,4,6,5)(2,9,7,10)(3,11,12,13)(8,15,14,16)(17,20,22,31)(18,24,26,38)(19,28,29,45)(21,35,33,32)(23,37,36,48)(25,42,40,39)(27,43,41,44)(30,47,34,46);(1,5,6,4)(2,10,7,9)(3,11,12,13)(8,15,14,16)

Polycyclic presentation:

2^10|b^2=ij,c^2=h,d^2=fi,e^2=fij,g^2=j,b^a=bf,c^a=ch,c^b=cg,d^a=di,d^b=dij,d^c=di,e^c=efij,f^c=fj,g^a=gi,g^c=gij,g^d=gj,h^b=hi,h^d=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319313, #72319918
maximal normal subgroup of	#72319313, #72319918

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