#72284954

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

2920 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:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=fgh,e^2=ghj,b^a=bf,c^a=cg,d^a=dj,d^b=dghi,d^c=dgi,e^a=egj,e^b=egh,e^c=efgij,e^d=ej,f^c=fi,f^d=fj,f^e=fij,g^b=gi,g^c=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285163, #72310940, #72310975, …
maximal normal subgroup of	#72285163, #72310940, #72310975, …

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