#72316887

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1439
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

3837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,d^2=ij,e^2=ij,f^2=j,h^2=i,b^a=bhj,c^a=cf,c^b=ce,d^a=dj,d^b=di,e^a=ei,e^c=eij,f^b=fij,f^c=fj,g^c=gij,h^b=hi,h^c=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316898, #72316910, #72316942
maximal normal subgroup of	#72316898, #72316910, #72316942

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