#72285474

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

2850 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)(3,13,15,8)(4,11)(5,10,14,16)(6,9)(7,12)(17,18)(19,27)(20,24,32,40)(21,33)(22,26)(23,30)(25,34)(28,38,43,36)(29,37)(31,39,44,45)(35,41)(42,48,46,47);(1,3)(2,8)(4,5)(6,14)(7,15)(9,10)(11,16)(12,13)(21,34)(23,27)(25,33)(31,44)(36,38)(37,41)(39,45)(47,48);(1,4)(2,9)(3,14,15,5)(6,7)(8,10,13,16)(11,12)(17,19)(18,23)(20,28,32,43)(21,35)(22,30)(24,38,40,36)(25,41)(26,27)(29,34)(31,42,44,46)(33,37)(39,48,45,47);(1,5)(2,10,12,16)(3,4)(6,15)(7,14)(8,9,13,11)(17,20)(18,24)(19,28,30,43)(21,31)(22,32)(23,36,27,38)(25,39)(26,40)(29,42,35,46)(33,45)(34,44)(37,47,41,48);(1,6,7,4)(2,11,12,9)(3,14,15,5)(8,10,13,16)(17,21,22,34)(18,25,26,33)(19,29,30,35)(20,31,32,44)(23,37,27,41)(24,39,40,45)(28,42,43,46)(36,47,38,48)

Polycyclic presentation:

2^10|a^2=f,c^2=f,d^2=h,e^2=i,b^a=bh,c^a=cfi,c^b=cg,d^a=dgij,d^c=dfghj,e^a=efi,e^b=ehij,e^c=efi,e^d=ej,f^b=fj,f^d=fi,g^a=gi,g^c=gj,g^d=gj,g^e=gi,h^a=hj,h^c=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185, #72340246, #72340258, …
maximal normal subgroup of	#72286185, #72340246, #72340258, …

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