#72322307

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂²×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2660 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,f^2=h,g^2=i,i^2=j,b^a=bi,c^b=ch,d^a=dj,d^b=dh,d^c=dh,e^a=ej,e^b=ehj,e^d=eh,f^a=fj,g^b=gj

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322319, #72322467
maximal normal subgroup of	#72322319, #72322467

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