#72340277

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

2791 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)(15,16)(17,18,23,28)(19,29,33,24)(20,32,35,25)(21,36,40,26)(22,27,41,30)(31,42,37,46)(34,43,38,44)(39,47,48,45);(1,2)(3,9,11,6)(4,12,14,7)(5,8)(10,16)(13,15)(17,19)(18,24,28,29)(20,27)(21,37,40,31)(22,25,41,32)(23,33)(26,42)(30,35)(34,47,38,45)(36,46)(39,44)(43,48);(1,3)(2,6)(4,13,14,10)(5,11)(7,16,12,15)(8,9)(17,20,23,35)(18,25)(19,30)(21,38,40,34)(22,24,41,29)(26,44)(27,33)(28,32)(31,45)(36,43)(37,47)(39,46,48,42);(4,14)(10,13)(17,21)(18,26)(19,31)(20,34)(22,39)(23,40)(24,42)(25,43)(27,45)(28,36)(29,46)(30,47)(32,44)(33,37)(35,38)(41,48);(1,4)(2,7,8,12)(3,10)(5,14)(6,15,9,16)(11,13)(17,22,23,41)(18,27,28,30)(19,32)(20,24)(21,39,40,48)(25,33)(26,45,36,47)(29,35)(31,43)(34,46)(37,44)(38,42)

Polycyclic presentation:

2^10|a^2=ij,b^2=g,c^2=ghij,e^2=fij,b^a=bh,c^a=cf,c^b=cfgh,d^a=dij,d^b=dgh,d^c=dij,e^b=efgi,e^c=ehij,e^d=efj,f^b=fi,f^c=fij,g^a=gij,g^c=gj,g^d=gij,g^e=gi,h^b=hij,h^c=hj,h^d=hij,h^e=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340284, #72340288, #72340301, …
maximal normal subgroup of	#72340284, #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, 1, F.9*F.10);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.9*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*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*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, 7, 1, F.7*F.9*F.10);;
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);;
gap> SetConjugate(coll, 8, 2, F.8*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);