#72284135

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

2839 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)(3,5)(4,13)(6,15)(7,9)(8,12)(10,16)(11,14)(18,23)(19,27)(20,30)(21,33)(32,34)(38,40)(39,45)(47,48);(1,3)(2,8)(4,14)(5,9)(6,11)(7,12)(10,15)(13,16)(17,18,22,23)(19,24,27,35)(20,25,30,36)(21,26,33,37)(28,38,31,40)(29,45,43,39)(32,41,34,42)(44,48,46,47);(1,4)(2,6)(3,10,12,14)(5,11,8,16)(7,15)(9,13)(17,19)(18,24)(20,31,30,28)(21,29)(22,27)(23,35)(25,40,36,38)(26,39)(32,46,34,44)(33,43)(37,45)(41,48,42,47);(1,5)(2,3)(4,11,15,16)(6,14,13,10)(7,8)(9,12)(17,20,22,30)(18,25,23,36)(19,28,27,31)(21,34)(24,38,35,40)(26,42)(29,46)(32,33)(37,41)(39,48)(43,44)(45,47);(1,6)(2,4)(3,11)(5,14)(7,13)(8,10)(9,15)(12,16)(17,21)(18,26)(19,29)(20,32)(22,33)(23,37)(24,39)(25,41)(27,43)(28,44)(30,34)(31,46)(35,45)(36,42)(38,47)(40,48)

Polycyclic presentation:

2^10|b^2=ij,c^2=f,d^2=hij,b^a=bi,c^a=ci,c^b=ch,d^a=dhij,d^b=dj,d^c=dg,e^a=efi,e^b=eg,e^d=efj,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709
maximal normal subgroup of	#72284709

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