#72319353

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#41×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4748 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:

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

Polycyclic presentation:

2^10|b^2=f,d^2=hj,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=cj,c^b=cg,d^b=dhj,d^c=dh,e^b=eh,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319381, #72319699, #72319749
maximal normal subgroup of	#72319381, #72319699, #72319749

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