#72318607

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	#8926850	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|c^2=fg,e^2=j,f^2=j,c^a=ch,c^b=cf,d^a=dfgh,d^b=dfghj,d^c=dfgh,e^a=efghj,e^b=ej,e^c=efghj,e^d=ej,f^a=fi,f^b=fj,f^c=fi,g^a=gi,g^b=gi,g^c=gi,g^f=gj,h^b=hij,h^f=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318725, #72319813, #72320178, …
maximal normal subgroup of	#72318725, #72319813, #72320178, …

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