#72284587

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	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2907 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)(3,10)(4,7)(5,14)(6,9)(8,16)(11,13)(12,15)(17,18)(19,23)(20,30)(21,25)(22,26)(24,33)(27,40)(28,45)(29,39)(31,37)(32,42)(34,38)(35,41)(36,43)(44,47)(46,48);(1,3,6,7)(2,4,9,10)(5,12,16,11)(8,15,14,13)(17,19)(18,23)(20,31)(21,34)(22,29)(24,40)(25,38)(26,39)(27,33)(28,36)(30,37)(32,44)(35,46)(41,48)(42,47)(43,45);(1,4)(2,7)(3,9)(5,15)(6,10)(8,12)(11,14)(13,16)(24,30)(27,31)(32,35)(47,48);(1,5)(2,8,9,14)(3,11)(4,13,10,15)(6,16)(7,12)(17,20)(18,24,26,30)(19,27)(21,35)(22,33)(23,37,39,40)(25,42,43,41)(28,46)(29,31)(32,36)(34,44)(38,48,45,47);(1,5)(2,8,9,14)(3,12)(4,13,10,15)(6,16)(7,11)(17,21)(18,25,26,43)(19,28)(20,32)(22,36)(23,38,39,45)(24,41,30,42)(27,44)(29,34)(31,46)(33,35)(37,47,40,48)

Polycyclic presentation:

2^10|b^2=ij,d^2=gh,e^2=gh,b^a=bij,c^a=cf,c^b=cf,d^a=df,d^b=di,d^c=dfg,e^a=efhj,e^b=eghj,e^c=ef,e^d=ej,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^d=gj,g^e=gj,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190
maximal normal subgroup of	#72285190

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