#72320879

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	C₂×#33675	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2424 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,f^2=h,g^2=i,i^2=j,b^a=bgi,c^b=cj,d^a=dhj,e^a=egh,e^d=eh,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321059, #72322669
maximal normal subgroup of	#72321059, #72322669

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