#72319362

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

4970 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,37)(27,36)(29,39)(30,40)(31,41)(32,43)(33,44)(34,42)(35,46)(38,45)(47,48);(2,7)(4,5)(8,14)(11,13)(17,19)(18,24)(20,31,33,47)(21,22)(23,30)(25,41,44,48)(26,28)(27,40)(29,32,34,38)(35,36)(37,46)(39,43,42,45);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,20)(18,25)(19,29,21,34)(22,33)(23,38,36,32)(24,39,28,42)(26,44)(27,45,37,43)(30,47)(31,35)(40,48)(41,46);(1,4,6,5)(2,9,7,10)(3,11,12,13)(8,15,14,16)(17,21,22,19)(18,24,26,28)(20,29,33,34)(23,30,36,35)(25,42,44,39)(27,46,37,40)(31,32,47,38)(41,45,48,43);(1,5,6,4)(2,10,7,9)(3,11,12,13)(8,15,14,16)

Polycyclic presentation:

2^10|b^2=ij,c^2=h,d^2=fij,e^2=fi,g^2=j,b^a=bf,c^a=ch,c^b=cg,d^a=dij,d^b=df,d^c=dfj,e^b=efi,f^c=fj,g^a=gi,g^c=gij,g^d=gj,h^b=hi,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	#72319411, #72319918
maximal normal subgroup of	#72319411, #72319918

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