#72284715

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2849 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)(3,4)(5,14)(6,10)(7,8)(9,16)(11,15)(12,13)(18,23)(19,27)(20,31)(21,35)(30,40)(34,45)(37,41)(43,44);(1,3)(2,7)(4,10)(5,11,16,12)(6,8)(9,15,14,13)(17,18)(19,28)(20,32)(21,36,35,26)(22,23)(24,27)(25,31)(29,41)(30,42,40,38)(33,37)(34,39,45,46)(43,48,44,47);(1,4,6,7)(2,8,10,3)(5,15)(9,12)(11,14)(13,16)(17,19,22,27)(18,24,23,28)(20,33,31,29)(21,30)(25,37,32,41)(26,38)(34,43)(35,40)(36,42)(39,48)(44,45)(46,47);(1,5,6,16)(2,9)(3,11)(4,13,7,15)(8,12)(10,14)(17,20)(18,25,23,32)(19,29,27,33)(21,34,35,45)(22,31)(24,37)(26,39)(28,41)(30,44)(36,46)(38,48,42,47)(40,43);(1,5)(2,9,10,14)(3,12)(4,13,7,15)(6,16)(8,11)(17,21,22,35)(18,26)(19,30,27,40)(20,34)(23,36)(24,38)(25,39,32,46)(28,42)(29,43)(31,45)(33,44)(37,47,41,48)

Polycyclic presentation:

2^10|b^2=f,c^2=fij,d^2=ghi,e^2=gj,b^a=bij,c^a=cfij,c^b=cfij,d^a=dfj,d^b=dfh,d^c=dg,e^a=efhj,e^b=eghj,e^c=efhij,e^d=ehi,f^d=fi,f^e=fij,g^a=gi,g^b=gj,g^c=gi,g^d=gj,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190, #72341033, #72341091, …
maximal normal subgroup of	#72285190, #72341033, #72341091, …

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