#72321835

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 ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2442 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)(4,12,14,8)(10,15,13,16)(17,18,21,23,20,27,33,36,22,25,34,26,32,39,46,41)(19,24,29,35,30,38,44,47,31,37,45,40,28,42,43,48);(1,3,5,9,6,11,2,7)(4,13,12,15,14,10,8,16)(18,23,25,26)(20,32)(21,33)(24,35,37,40)(27,41,39,36)(28,30)(29,44)(34,46)(38,48,42,47)(43,45);(17,19)(18,24)(20,30)(21,29)(22,31)(23,35)(25,37)(26,40)(27,38)(28,32)(33,44)(34,45)(36,47)(39,42)(41,48)(43,46);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,16)(11,13)(18,25)(23,26)(24,37)(27,39)(35,40)(36,41)(38,42)(47,48);(3,11)(4,14)(7,9)(8,12)(17,20)(18,26)(19,28)(22,32)(23,25)(24,35)(27,36)(29,45)(30,31)(33,46)(37,40)(38,48)(39,41)(42,47)

Polycyclic presentation:

2^10|a^2=g,b^2=f,f^2=h,g^2=i,i^2=j,b^a=bgij,d^a=dhj,d^b=dh,e^a=eghj,e^b=ehj,e^c=ej,e^d=eh,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321949, #72322376
maximal normal subgroup of	#72321949, #72322376

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