#72285221

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

5843 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=g,e^2=gij,c^a=cf,c^b=cij,d^a=dfi,d^b=di,d^c=dfg,e^a=efh,e^b=egij,e^c=efhj,e^d=ei,f^d=fi,f^e=fij,g^a=gi,g^c=gi,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

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