#72341238

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	#35527	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2717 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=j,d^2=i,e^2=ij,b^a=bf,c^a=cfij,c^b=cf,d^a=di,d^b=dg,d^c=dh,e^a=eghij,e^b=eij,e^c=ehj,f^d=fi,f^e=fij,g^a=gi,g^c=gj,h^a=hi,h^b=hij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341241
maximal normal subgroup of	#72341241

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