#72340213

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	#245252	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2796 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fghj,e^2=j,b^a=bg,c^a=cgh,c^b=cfg,d^a=di,d^b=di,d^c=dfi,e^b=ehj,e^c=egj,e^d=egj,f^a=fi,f^c=fi,f^e=fj,g^b=gi,g^c=gj,h^b=hj,h^c=hij,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340228, #72340248, #72340378
maximal normal subgroup of	#72340228, #72340248, #72340378

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