#72286800

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	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=j,c^2=f,d^2=g,f^2=h,g^2=i,i^2=j,c^a=ch,c^b=chj,d^a=dh,d^b=dhj,d^c=dg,e^a=eh,e^b=eh,e^d=eg,f^d=fj,g^c=gi,g^e=gij,i^c=ij,i^e=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287099, #72287278, #72287657, …
maximal normal subgroup of	#72287099, #72287278, #72287657, …

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