#72287814

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, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2330 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)(3,8)(4,14)(5,10)(6,9)(7,11)(12,15)(13,16);(1,3,6,12,7,13,4,5)(2,8,9,15,11,16,14,10)(17,18)(19,29,31,43)(20,33,30,46)(21,36)(22,27)(23,28)(24,39,41,32)(25,44,42,35)(26,37)(34,47)(38,40)(45,48);(1,4,7,6)(2,9,11,14)(3,5,13,12)(8,15,16,10)(17,19,21,20,22,32,34,35,23,31,37,30,38,39,48,44)(18,24,26,42,27,43,45,33,28,41,36,25,40,29,47,46);(1,5,4,13,7,12,6,3)(2,10,14,16,11,15,9,8)(19,30,31,20)(21,34)(22,38)(24,25,41,42)(26,45)(27,40)(29,46,43,33)(32,35,39,44)(36,47)(37,48);(1,4,7,6)(2,9,11,14)(3,12,13,5)(8,10,16,15)(17,20,34,31,38,44,21,32,23,30,48,19,22,35,37,39)(18,25,45,24,40,33,26,29,28,42,47,41,27,46,36,43)

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=fh,e^2=gi,f^2=h,g^2=i,i^2=j,c^a=ch,c^b=cij,d^c=dghi,e^a=eh,e^b=ehij,e^d=egij,f^c=fj,f^e=fj,g^b=gj,g^d=gi,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287829, #72319058, #72319265
maximal normal subgroup of	#72287829, #72319058, #72319265

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