#72286969

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	#7624082	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2206 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fhi,c^2=ghj,f^2=h,g^2=i,i^2=j,b^a=bg,c^a=cghj,c^b=cghj,d^b=dj,d^c=dhj,e^b=eghj,e^c=eghj,f^b=fj,f^c=fj,g^a=gi,g^b=gi,g^e=gij,i^a=ij,i^b=ij,i^e=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287030, #72287751, #72287809, …
maximal normal subgroup of	#72287030, #72287751, #72287809, …
maximal quotient of	#72343708, #72343766

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