#72287424

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	#489290	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #17320 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1241 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=e,c^2=efhi,d^2=e,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bg,c^a=cgij,c^b=cf,d^a=dij,d^b=di,d^c=dhi,e^c=ej,f^b=fh,f^c=fj,f^d=fj,g^a=gij,g^b=gij,g^c=gij,h^b=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.9);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.6*F.8*F.9);;
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, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9*F.10);;
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);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
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, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);