#72286718

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×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2474 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)(4,14)(5,10)(6,7)(8,13)(9,12)(15,16);(1,2)(3,11)(4,14)(5,10)(6,7)(8,13)(9,12)(15,16)(17,18)(19,25)(20,34)(21,26)(22,27)(23,28)(24,29)(30,44)(31,40)(32,41)(33,42)(35,47)(36,45)(37,43)(38,48)(39,46);(2,7)(8,11)(9,14)(10,16)(17,19,22,31,39,38,21,37,24,20,35,36,23,33,30,32)(18,25,27,40,46,48,26,43,29,34,47,45,28,42,44,41);(1,3,5,12,6,13,15,4)(2,8,10,14,7,11,16,9)(17,20,21,37,23,38,22,31,24,19,30,32,39,33,35,36)(18,25,26,41,28,42,27,45,29,34,44,43,46,48,47,40);(1,4,15,13,6,12,5,3)(2,9,16,11,7,14,10,8)(19,31,20,36)(21,22)(23,39)(25,40,34,45)(26,27)(28,46)(30,35)(32,33,37,38)(41,42,43,48)(44,47)

Polycyclic presentation:

2^10|c^2=g,d^2=f,e^2=fghi,f^2=h,g^2=ij,h^2=j,i^2=j,c^a=chij,c^b=chij,d^a=dhij,d^b=dhi,d^c=di,e^a=ehij,e^b=ehij,e^c=egi,e^d=egi,f^c=fj,f^e=fi,g^d=gj,g^e=gij,h^e=hj,i^e=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287099, #72287390, #72287751, …
maximal normal subgroup of	#72287099, #72287390, #72287751, …

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