#72284299

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2913 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)(3,9)(4,8)(5,7)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,34)(22,27)(24,30)(26,36)(28,37)(29,38)(31,39)(32,41)(33,42)(35,40)(43,46)(44,48)(45,47);(1,3)(2,6)(4,12,13,10)(5,11)(7,9)(8,15,16,14)(17,19)(18,23)(20,28,33,31)(21,35)(22,30)(24,27)(25,37,42,39)(26,38)(29,36)(32,45,46,44)(34,40)(41,47,43,48);(2,7)(4,13)(6,9)(8,16)(17,19)(18,24)(20,31)(21,35)(22,30)(23,27)(25,39)(26,40)(28,33)(29,36)(32,45)(34,38)(37,42)(41,47)(43,48)(44,46);(1,4)(2,8,7,16)(3,10)(5,13)(6,14,9,15)(11,12)(17,20)(18,25,27,42)(19,28,30,31)(21,32,36,46)(22,33)(23,37)(24,39)(26,43)(29,44)(34,41)(35,45)(38,48,40,47);(1,4)(2,8)(3,10,11,12)(5,13)(6,15,9,14)(7,16)(17,21,22,36)(18,26,27,34)(19,29)(20,32)(23,38)(24,40)(25,41)(28,44,31,45)(30,35)(33,46)(37,47,39,48)(42,43)

Polycyclic presentation:

2^10|b^2=f,d^2=ghij,e^2=gj,b^a=bij,c^a=cfi,c^b=cf,d^a=dfi,d^b=dg,d^c=dh,e^a=egj,e^b=efh,e^c=eh,e^d=ehij,f^d=fi,f^e=fij,g^b=gi,g^c=gj,g^d=gj,h^a=hi,h^b=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284400, #72285132, #72310969, …
maximal normal subgroup of	#72284400, #72285132, #72310969, …

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