#72340249

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
prime radical	#245249	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2800 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=j,e^2=fj,b^a=bg,c^a=cg,c^b=cfgh,d^a=di,d^b=di,d^c=dgi,e^a=ehj,e^b=ei,e^c=efghj,e^d=efij,f^a=fi,f^c=fij,g^b=gi,g^e=gj,h^b=hj,h^c=hij,h^d=hi,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340254, #72340288, #72340301, …
maximal normal subgroup of	#72340254, #72340288, #72340301, …

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