#72319216

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	#7703033	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4828 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=i,c^2=hij,d^2=fij,e^2=fi,g^2=j,b^a=bfi,c^a=ch,c^b=cg,d^a=dfij,e^c=efi,f^c=fj,g^a=gi,g^c=gi,h^a=hj,h^b=hij,h^c=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

82 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, 2, F.9);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*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*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, 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);