#72284197

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂×#117	C₂⁴
prime radical	C₂×#23988	C₂
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

3537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,e^2=j,f^2=i,b^a=bf,c^a=cefij,c^b=cg,d^a=dij,d^b=dj,d^c=dj,e^b=ei,e^c=eij,f^b=fh,g^a=ghi,g^f=gj,h^b=hj,h^c=hj

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284412, #72284911, #72285471
maximal normal subgroup of	#72284412, #72284911, #72285471

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