#72285673

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1864
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×#135	Q₈
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

1360 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=e,c^2=ghj,e^2=i,f^2=h,g^2=i,h^2=j,c^a=cg,c^b=cf,d^a=dij,d^b=di,d^c=di,e^c=ej,f^a=fi,f^b=fh,f^c=fhi,g^a=gij,g^b=gi,g^c=gj,h^b=hj,h^c=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285790
maximal normal subgroup of	#72285790

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