#72287564

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#106
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#135	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#135 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

729 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=g,c^2=e,d^2=eghi,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bg,c^b=cgi,d^a=dg,d^b=di,d^c=df,e^d=ej,f^a=fij,f^c=fh,f^d=fhij,g^a=gij,g^c=gij,h^c=hj,h^d=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287651, #72287720
maximal normal subgroup of	#72287651, #72287720

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