#72303279

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1436
commutator	C₂³×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
socle	C₂²	#7557
prime radical	C₂⁵	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

2407 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=d,c^2=d,g^2=i,h^2=j,b^a=bg,c^a=cf,c^b=cde,d^a=di,e^a=eij,f^b=fj,f^c=fi,h^b=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303319
maximal normal subgroup of	#72303319

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