#72285585

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	#251860	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2731 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=i,c^2=fg,e^2=j,b^a=bfg,c^a=cfh,d^a=dhij,d^b=dfgi,d^c=dfgj,e^a=ej,e^b=efhj,e^c=efi,e^d=ej,f^b=fi,f^c=fj,f^d=fj,g^a=gi,g^b=gi,g^c=gj,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285772, #72340307, #72340327, …
maximal normal subgroup of	#72285772, #72340307, #72340327, …

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