#72286239

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

2850 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=gh,e^2=gij,b^a=bh,c^a=cfg,c^b=ch,d^a=dfij,d^b=dhij,d^c=dfij,e^a=ehj,e^b=efgi,e^c=efj,e^d=ehij,f^b=fi,f^c=fj,f^e=fi,g^a=gj,g^b=gi,g^c=gi,g^d=gj,h^a=hi,h^c=hi,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340254, #72340293, #72340307, …
maximal normal subgroup of	#72340254, #72340293, #72340307, …

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