#72287448

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21452
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#61436	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #17320 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=g,d^2=e,e^2=i,f^2=h,g^2=ij,h^2=j,c^a=cg,c^b=cf,d^a=dij,d^b=di,d^c=dhij,e^c=ej,f^a=fij,f^b=fh,f^c=fhi,f^d=fj,g^a=gij,g^b=gij,h^b=hj,h^c=hj

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287455
maximal normal subgroup of	#72287455

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