#72319053

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	#7623523	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2356 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)(3,11)(4,10)(13,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,34)(28,38)(29,36)(30,37)(31,41)(32,39)(33,40)(35,44)(42,47)(43,46)(45,48);(1,2,6,10,7,8,5,4)(3,12,13,15,11,16,14,9)(17,19,20,30,33,44,45,42,22,29,32,43,21,28,31,27)(18,23,25,37,40,35,48,47,24,36,39,46,26,38,41,34);(1,3)(2,9)(4,12)(5,13)(6,14)(7,11)(8,15)(10,16)(19,27)(20,31)(21,33)(23,34)(25,41)(26,40)(28,30)(29,42)(32,45)(35,46)(36,47)(37,38)(39,48)(43,44);(1,4,5,8,7,10,6,2)(3,12,13,15,11,16,14,9)(18,24)(19,28,29,44)(20,32)(23,35,36,38)(26,40)(27,30,42,43)(31,45)(34,46,47,37);(2,8)(4,10)(9,15)(12,16)(19,29)(23,36)(27,42)(28,44)(30,43)(34,47)(35,38)(37,46)

Polycyclic presentation:

2^10|b^2=g,d^2=f,f^2=h,g^2=ij,i^2=j,b^a=bh,c^a=ch,c^b=cg,d^a=dhj,d^b=dhij,e^b=ehj,e^d=eh,f^b=fj,g^c=gi,g^d=gj,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319103, #72319621
maximal normal subgroup of	#72319103, #72319621

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