#72286151

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2793 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)(3,9)(4,11)(5,13)(6,8)(7,16)(10,12)(14,15)(20,30)(21,33)(27,42)(28,44);(1,3)(2,4,8,9)(5,14)(6,11)(7,15,13,12)(10,16)(17,18)(19,23,29,37)(20,31)(21,25,33,41)(22,26)(24,30)(27,38,42,35)(28,36)(32,40,34,39)(43,48)(44,47)(45,46);(1,4)(2,3)(5,15,16,12)(6,9)(7,14,13,10)(8,11)(17,19)(18,23,26,37)(20,27)(21,28)(22,29)(24,38,31,35)(25,36,41,47)(30,42)(32,43)(33,44)(34,45)(39,48,40,46);(1,5)(2,7,8,13)(3,10)(4,12,9,15)(6,16)(11,14)(17,20)(18,24)(19,27,29,42)(21,34,33,32)(22,30)(23,35,37,38)(25,40,41,39)(26,31)(28,43)(36,46)(44,45)(47,48);(2,8)(4,9)(10,14)(12,15)(17,21,22,33)(18,25,26,41)(19,28,29,44)(20,32,30,34)(23,36,37,47)(24,39,31,40)(27,43,42,45)(35,46,38,48)

Polycyclic presentation:

2^10|b^2=f,c^2=fgh,d^2=f,e^2=ij,b^a=bh,c^a=cj,c^b=cg,d^a=dfghij,d^b=di,d^c=dfgi,e^a=efghi,e^b=efghij,e^c=ej,e^d=efh,f^a=fj,f^c=fi,f^e=fij,g^a=gj,g^b=gi,g^c=gij,g^d=gi,h^b=hj,h^c=hj,h^d=hij,h^e=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185, #72340295, #72340301, …
maximal normal subgroup of	#72286185, #72340295, #72340301, …

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