#72340226

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

2852 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)(6,9)(7,13)(10,16)(17,18)(19,23)(20,24)(21,34)(22,26)(25,35)(27,36)(28,38)(29,37)(30,39)(31,41)(32,46)(33,42)(40,45)(43,47)(44,48);(1,2)(3,9,11,6)(4,13)(5,8)(7,12)(10,14,16,15)(17,19)(18,23)(20,27,33,45)(21,29)(22,30)(24,40,42,36)(25,38)(26,39)(28,35)(31,43,32,44)(34,37)(41,48,46,47);(1,2)(3,6)(4,13,12,7)(5,8)(9,11)(10,14,16,15)(17,20)(18,24,26,42)(19,27,30,45)(21,32)(22,33)(23,36)(25,41,34,46)(28,43,29,44)(31,35)(37,48)(38,47)(39,40);(1,3)(2,6)(4,10,12,16)(5,11)(7,14,13,15)(8,9)(17,21)(18,25,26,34)(19,28,30,29)(20,31)(22,35)(23,37)(24,41,42,46)(27,43,45,44)(32,33)(36,47)(38,39)(40,48);(1,4,5,12)(2,7,8,13)(3,10)(6,14)(9,15)(11,16)(17,21)(18,25)(19,29)(20,32,33,31)(22,35)(23,38)(24,41,42,46)(26,34)(27,44,45,43)(28,30)(36,47,40,48)(37,39)

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=g,e^2=fj,b^a=bf,c^a=cfi,c^b=cfg,d^a=dh,d^b=dh,d^c=dgi,e^a=eij,e^b=ej,e^c=efhi,e^d=efg,f^c=fi,f^d=fj,g^a=gi,g^b=gi,g^e=gj,h^b=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, #72340284, #72340311, …
maximal normal subgroup of	#72340228, #72340284, #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, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.6*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
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);