#72285414

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

5992 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:

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

Polycyclic presentation:

2^10|c^2=f,e^2=i,c^a=cg,c^b=ch,d^a=dj,d^b=dgj,e^a=eij,e^b=ehij,e^c=ef,e^d=ei,f^a=fi,f^b=fj,g^c=gi,g^e=gi,h^c=hj,h^d=hi

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185
maximal normal subgroup of	#72286185

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