#72283990

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	#251828	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2721 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(21,33)(24,30)(28,42)(35,39);(1,3)(2,7,9,10)(4,12)(5,11)(6,13)(8,15,14,16)(17,18)(19,23,29,37)(20,30,32,24)(21,34)(22,26)(25,33)(27,39)(28,43,42,36)(31,38,46,45)(35,41)(40,47)(44,48);(1,4,6,5)(2,8,9,14)(3,11)(7,15)(10,16)(12,13)(17,19,22,29)(18,23)(20,27,32,41)(21,28)(24,35)(25,36,34,43)(26,37)(30,39)(31,44)(33,42)(38,48,45,47)(40,46);(1,5,6,4)(2,8,9,14)(3,12)(7,16)(10,15)(11,13)(17,20)(18,24,26,30)(19,27)(21,31,33,46)(22,32)(23,35,37,39)(25,38)(28,44,42,40)(29,41)(34,45)(36,48)(43,47);(4,5)(11,12)(17,21,22,33)(18,25,26,34)(19,28,29,42)(20,31,32,46)(23,36,37,43)(24,38,30,45)(27,40,41,44)(35,47,39,48)

Polycyclic presentation:

2^10|b^2=h,c^2=ghij,d^2=ghi,e^2=j,b^a=bf,c^b=cfg,d^a=dghi,d^b=dfghj,e^a=efghj,e^b=ej,e^c=ehij,e^d=ehij,f^b=fi,f^c=fi,f^d=fij,f^e=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gj,g^e=gj,h^a=hi,h^c=hj,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284081, #72284129, #72285526, …
maximal normal subgroup of	#72284081, #72284129, #72285526, …

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