#72284538

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

2838 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:

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

Polycyclic presentation:

2^10|b^2=fij,d^2=ghi,e^2=gi,b^a=bf,c^a=cf,c^b=cj,d^a=dfi,d^b=dfg,d^c=dgh,e^a=efhi,e^b=efghij,e^c=ef,e^d=eh,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gij,g^d=gj,h^b=hj,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	#72284848, #72310969, #72340293, …
maximal normal subgroup of	#72284848, #72310969, #72340293, …

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