#72341101

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#6432958	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5851 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)(3,7)(4,8,12,16)(5,13)(6,9)(10,15,11,14)(17,18)(19,23,29,38)(20,30)(21,26)(22,25)(24,32)(27,36,31,39)(28,37,44,47)(33,45)(34,40)(35,41)(42,46,43,48);(1,3,6,5)(2,7,9,13)(4,11,12,10)(8,15,16,14)(17,19,22,29)(18,23)(20,31)(21,28,35,44)(24,39,30,36)(25,38)(26,37)(27,32)(33,43)(34,42)(40,48,45,46)(41,47);(1,4)(2,8,9,16)(3,10,5,11)(6,12)(7,14)(13,15)(17,20,22,32)(18,24)(19,27,29,31)(21,34)(23,36)(25,30)(26,40,41,45)(28,43)(33,35)(37,46,47,48)(38,39)(42,44);(1,5,6,3)(2,7,9,13)(4,11,12,10)(8,14,16,15)(18,25)(20,32)(26,41)(33,34);(4,12)(8,16)(10,11)(14,15)(17,21,22,35)(18,26,25,41)(19,28,29,44)(20,33,32,34)(23,37,38,47)(24,40,30,45)(27,42,31,43)(36,46,39,48)

Polycyclic presentation:

2^10|a^2=g,b^2=hi,c^2=fghij,d^2=j,e^2=ij,b^a=bgh,c^a=cf,c^b=cg,d^a=dgi,d^b=dh,d^c=dgi,e^c=ehj,f^a=fj,f^b=fi,f^c=fi,f^d=fj,f^e=fij,g^b=gij,g^c=gj,h^a=hij,h^c=hij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341103, #72341169
maximal normal subgroup of	#72341103, #72341169

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