#72284912

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

2797 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)(4,12)(5,8)(6,11)(7,14)(10,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,34)(28,40)(29,36)(30,45)(31,38)(32,35)(33,41)(37,43)(39,46)(42,47)(44,48);(1,2)(3,6)(4,7,14,12)(5,8)(9,11)(10,16,13,15)(17,19)(18,23)(20,27,31,43)(21,32,33,28)(22,29)(24,37,38,34)(25,40,41,35)(26,36)(30,42)(39,48)(44,46)(45,47);(1,3)(2,6)(4,13)(5,11)(7,15)(8,9)(10,14)(12,16)(24,38)(25,41)(34,37)(35,40);(1,4,5,14)(2,7)(3,10)(6,15,9,16)(8,12)(11,13)(17,20)(18,24,26,38)(19,27,29,43)(21,30,33,46)(22,31)(23,34)(25,39)(28,44)(32,42)(35,48,40,47)(36,37)(41,45);(2,8)(3,11)(4,14)(6,9)(10,13)(15,16)(17,21)(18,25)(19,28)(20,30)(22,33)(23,35)(24,39)(26,41)(27,42)(29,32)(31,46)(34,47)(36,40)(37,48)(38,45)(43,44)

Polycyclic presentation:

2^10|b^2=f,d^2=gij,b^a=bf,c^a=cfij,d^a=dfj,d^b=dfg,d^c=dfgh,e^a=ehij,e^b=ehij,e^c=efghij,e^d=ehi,f^d=fi,f^e=fj,g^a=gi,g^b=gi,g^c=gij,g^e=gj,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190, #72310940, #72311043, …
maximal normal subgroup of	#72285190, #72310940, #72311043, …

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