#72286998

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,i^2=j,b^a=bj,c^b=cg,d^b=dj,d^c=dh,e^a=ej,e^b=ehij,e^c=egj,e^d=ej,f^c=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287030, #72320823
maximal normal subgroup of	#72287030, #72320823

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