#72265732

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1893
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#28143	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2287 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:

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

Polycyclic presentation:

2^10|c^2=f,e^2=hi,f^2=i,i^2=j,b^a=bg,c^b=ce,d^c=dj,e^a=eh,e^b=ehij,f^b=fhi,g^c=gh,i^b=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265733, #72308543, #72308636
maximal normal subgroup of	#72265733, #72308543, #72308636

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