#72340221

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

2968 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,36)(23,27)(28,41)(29,38)(30,40)(31,37)(32,42)(33,44)(34,39)(35,43)(45,48)(46,47);(6,9)(7,12)(17,19)(18,24)(20,31)(21,34)(22,29)(23,30)(25,41)(26,39)(27,40)(28,33)(32,46)(35,45)(36,38)(37,44)(42,48)(43,47);(1,2)(3,9)(4,12,13,7)(5,8)(6,11)(10,15,16,14)(17,20,23,33)(18,25,27,44)(19,28)(21,35,22,32)(24,37)(26,43,36,42)(29,45)(30,31)(34,46)(38,47)(39,48)(40,41);(1,3)(2,6)(4,10,13,16)(5,11)(7,14,12,15)(8,9)(17,21)(18,26)(19,29,30,34)(20,32)(22,23)(24,38,40,39)(25,42)(27,36)(28,45,31,46)(33,35)(37,47,41,48)(43,44);(1,4,5,13)(2,7,8,12)(3,10)(6,14)(9,15)(11,16)(17,22,23,21)(18,26,27,36)(19,29)(20,32)(24,39)(25,43)(28,46,31,45)(30,34)(33,35)(37,47,41,48)(38,40)(42,44)

Polycyclic presentation:

2^10|c^2=gi,d^2=g,e^2=fgij,b^a=bf,c^a=cg,c^b=cf,d^b=dh,d^c=dgij,e^a=ei,e^b=ehi,e^c=efghi,e^d=efi,f^c=fi,f^d=fj,f^e=fj,g^b=gi,g^e=gj,h^a=hj,h^c=hij,h^d=hi,h^e=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340228, #72340239, #72340311, …
maximal normal subgroup of	#72340228, #72340239, #72340311, …

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