#72340281

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

2791 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=gi,d^2=g,e^2=ij,b^a=bi,c^a=cg,c^b=cf,d^a=di,d^b=dgh,d^c=dij,e^a=eh,e^b=efi,e^c=ehj,e^d=ef,f^a=fi,f^c=fi,f^d=fj,g^b=gi,g^e=gj,h^b=hi,h^c=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340288, #72340301, #72340305, …
maximal normal subgroup of	#72340288, #72340301, #72340305, …

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