#72340219

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

2798 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)(3,9,11,6)(4,7,12,14)(5,8)(10,15)(13,16)(17,18,22,26)(19,27)(20,31)(21,35)(23,30)(24,34)(25,37)(28,42,32,38)(29,43,46,39)(33,41,36,40)(44,47)(45,48);(6,9)(10,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,38)(25,39)(26,27)(28,34)(31,42)(33,45)(35,43)(36,44)(37,46)(40,47)(41,48);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,20)(18,24)(19,28,30,32)(21,36)(22,34)(23,38,27,42)(25,41)(26,31)(29,45,46,44)(33,37)(35,40)(39,48,43,47);(4,12)(10,16)(17,21)(18,25)(19,29)(20,33)(22,37)(23,39)(24,40)(26,35)(27,43)(28,44)(30,46)(31,41)(32,45)(34,36)(38,47)(42,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,30)(28,32)(39,43)(47,48)

Polycyclic presentation:

2^10|a^2=f,c^2=ghj,e^2=j,b^a=bh,c^a=cg,c^b=cfgij,d^a=dfhi,d^c=di,e^a=egh,e^b=ehi,e^d=egh,f^b=fi,f^c=fij,f^d=fi,f^e=fj,g^a=gij,g^b=gi,g^c=gj,g^d=gi,h^a=hi,h^c=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340228, #72340288, #72340329, …
maximal normal subgroup of	#72340228, #72340288, #72340329, …

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