#72286716

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 ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2375 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:

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

Polycyclic presentation:

2^10|b^2=g,d^2=f,e^2=fgh,f^2=h,g^2=i,h^2=j,i^2=j,c^b=chi,d^a=dj,d^b=dij,d^c=dj,e^b=egij,e^d=eg,f^b=fj,f^e=fi,g^d=gj,g^e=gi,h^e=hj,i^e=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286877, #72287030, #72287790, …
maximal normal subgroup of	#72286877, #72287030, #72287790, …

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