#72285545

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

5994 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)(3,12,5,8)(4,9,14,16)(6,15)(7,11)(10,13)(17,18)(19,23,29,38)(20,30,31,24)(21,33)(22,26)(25,34)(27,39)(28,37,44,46)(32,41,35,40)(36,42)(43,47)(45,48);(1,3)(2,8)(4,6)(5,7)(9,15)(10,16)(11,12)(13,14)(20,31)(21,34)(24,30)(27,42)(32,35)(36,39)(37,46)(47,48);(1,4)(2,9,11,16)(3,13,5,6)(7,14)(8,15)(10,12)(17,19)(18,23,26,38)(20,27)(21,28,34,44)(22,29)(24,39,30,36)(25,37)(31,42)(32,45,35,43)(33,46)(40,47)(41,48);(1,5,7,3)(2,8)(4,6)(9,15,16,10)(11,12)(13,14)(17,20,22,31)(18,24,26,30)(19,27,29,42)(21,35)(23,36,38,39)(25,41)(28,43)(32,34)(33,40)(37,47)(44,45)(46,48);(1,6)(2,10)(3,14)(4,5)(7,13)(8,9)(11,15)(12,16)(17,21)(18,25)(19,28)(20,32)(22,34)(23,37)(24,40)(26,33)(27,43)(29,44)(30,41)(31,35)(36,47)(38,46)(39,48)(42,45)

Polycyclic presentation:

2^10|a^2=f,c^2=fg,d^2=hi,b^a=bh,c^a=cg,c^b=ch,d^a=dfg,d^b=dhi,d^c=dfgh,e^a=efi,e^b=eghi,e^c=efij,e^d=efj,f^b=fj,f^c=fi,f^d=fi,g^a=gi,g^c=gi,g^d=gij,g^e=gi,h^a=hj,h^c=hj,h^e=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, 1, F.6);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
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.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*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);