#72284263

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	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|c^2=ghi,e^2=ghij,c^a=ch,c^b=cg,d^b=df,d^c=dfhi,e^a=ef,e^b=ef,e^c=eghij,e^d=ehij,f^c=fi,f^e=fij,g^a=gij,g^c=gij,g^d=gj,g^e=gj,h^b=hij,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284848, #72310940, #72340239, …
maximal normal subgroup of	#72284848, #72310940, #72340239, …

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