#72319202

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	#6453848	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=hij,d^2=fi,e^2=fij,g^2=j,b^a=bi,c^a=ch,c^b=cg,d^a=df,d^b=di,d^c=dfj,e^a=efij,f^c=fj,g^a=gi,g^b=gj,g^c=gij,g^d=gj,h^a=hj,h^b=hi,h^c=hj,h^d=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319313
maximal normal subgroup of	#72319313

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