#72284813

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

2877 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)(5,9)(8,12)(10,14)(11,15)(13,16)(19,27)(20,30)(21,33)(28,31)(29,42)(36,43)(37,41)(44,45);(1,3)(2,6)(4,10,12,16)(5,11)(7,13,8,14)(9,15)(17,18)(19,23,27,38)(20,24,30,40)(21,34,33,25)(22,26)(28,43)(29,37)(31,36)(32,46)(35,39)(41,42)(44,48,45,47);(1,4)(2,7)(3,10,11,16)(5,12)(6,13,15,14)(8,9)(17,19,22,27)(18,23)(20,31)(21,29)(24,36,40,43)(25,41,34,37)(26,38)(28,30)(32,45,35,44)(33,42)(39,48)(46,47);(3,11)(7,8)(10,16)(13,14)(17,20,22,30)(18,24,26,40)(19,28,27,31)(21,35,33,32)(23,36,38,43)(25,39,34,46)(29,44,42,45)(37,48,41,47);(1,4)(2,8,9,7)(3,10,11,16)(5,12)(6,14)(13,15)(17,21)(18,25)(19,29,27,42)(20,32,30,35)(22,33)(23,37,38,41)(24,39,40,46)(26,34)(28,44)(31,45)(36,47)(43,48)

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=j,e^2=ghij,b^a=bf,c^a=cg,c^b=cfgj,d^a=dfg,d^b=dh,d^c=dhi,e^a=efg,e^b=efgj,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

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284859, #72284923, #72340239, …
maximal normal subgroup of	#72284859, #72284923, #72340239, …

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