#72287375

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	#7623243	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2246 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=f,f^2=h,g^2=i,i^2=j,d^a=dhj,d^b=dhj,d^c=dghi,e^a=ej,e^b=ehj,e^c=eg,f^c=fj,g^d=gi,g^e=gij,i^d=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287431, #72318539, #72318603
maximal normal subgroup of	#72287431, #72318539, #72318603

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