#72287503

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	#7623517	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2302 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=g,e^2=fh,f^2=h,g^2=i,i^2=j,c^a=chj,c^b=chj,d^a=dhj,d^b=dhj,d^c=dhi,e^a=ej,e^b=ehj,e^d=egi,f^d=fj,g^c=gj,g^e=gi,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287517, #72318603, #72318707
maximal normal subgroup of	#72287517, #72318603, #72318707

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