#72287808

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	#33520	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2514 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=fhj,e^2=fi,f^2=h,g^2=i,i^2=j,b^a=bhj,c^a=chj,c^b=cghij,d^a=dhj,d^b=dgij,d^c=dh,e^b=eij,e^c=eij,e^d=eij,f^b=fj,g^c=gi,g^d=gi,g^e=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287829, #72318999, #72319510
maximal normal subgroup of	#72287829, #72318999, #72319510

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