#72322270

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1879
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#7623523	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=e,c^2=eg,f^2=hj,g^2=i,h^2=j,b^a=bf,c^a=chij,c^b=cfgi,d^b=dij,e^a=ei,e^c=ej,f^a=fh,f^b=fhi,f^c=fhi,g^a=gi,g^b=gj,g^c=gj,h^a=hj,h^b=hj,h^c=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322289
maximal normal subgroup of	#72322289

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