#72340273

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

2847 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=j,c^2=ghj,e^2=ghj,b^a=bh,c^a=cf,c^b=cfg,d^a=dij,d^b=dfghi,d^c=dij,e^a=eh,e^b=efgj,e^d=eghj,f^b=fi,f^c=fij,f^e=fj,g^a=gij,g^b=gi,g^c=gj,g^d=gij,g^e=gij,h^c=hj,h^d=hij,h^e=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340284, #72340293, #72340295, …
maximal normal subgroup of	#72340284, #72340293, #72340295, …

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