#72284701

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
prime radical	#251827	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2777 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(19,27)(24,38)(25,40)(44,45);(1,3,6,12)(2,7,10,16)(4,11)(5,13)(8,14)(9,15)(17,18)(19,23,27,37)(20,24,32,38)(21,25,34,40)(22,26)(28,43)(29,41)(30,35)(31,39)(33,46)(36,42)(44,47,45,48);(1,4,6,5)(2,8,10,9)(3,11)(7,14)(12,13)(15,16)(17,19,22,27)(18,23)(20,30)(21,29)(24,35,38,43)(25,41,40,36)(26,37)(28,32)(31,45,33,44)(34,42)(39,48)(46,47);(3,12)(8,9)(11,13)(14,15)(17,20,22,32)(18,24,26,38)(19,28,27,30)(21,33,34,31)(23,35,37,43)(25,39,40,46)(29,44,42,45)(36,48,41,47);(1,5,6,4)(2,9)(3,13)(7,15,16,14)(8,10)(11,12)(17,21)(18,25)(19,29,27,42)(20,31,32,33)(22,34)(23,36,37,41)(24,39,38,46)(26,40)(28,44)(30,45)(35,47)(43,48)

Polycyclic presentation:

2^10|b^2=fij,c^2=g,d^2=j,e^2=ghij,b^a=bf,c^a=cg,c^b=cfgi,d^a=dfgi,d^b=dh,d^c=dhi,e^a=efgi,e^b=efgij,e^c=ehi,e^d=ehi,f^c=fi,f^d=fj,f^e=fi,g^b=gi,g^d=gj,g^e=gj,h^a=hij,h^b=hj,h^c=hj,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284743, #72284775, #72285772, …
maximal normal subgroup of	#72284743, #72284775, #72285772, …

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