#72284586

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

2830 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)(3,7)(4,11)(5,6)(8,13)(9,15)(10,14)(12,16)(17,18)(19,23)(20,26)(21,25)(22,27)(24,33)(28,39)(29,38)(30,40)(31,37)(32,41)(34,46)(35,42)(36,43)(44,47)(45,48);(1,3,6,10)(2,7,5,14)(4,12,13,9)(8,16,11,15)(17,19,22,30)(18,23,27,40)(20,31,33,28)(21,34,36,29)(24,39,26,37)(25,38,43,46)(32,45,35,44)(41,47,42,48);(1,4)(2,8,5,11)(3,9,10,12)(6,13)(7,15)(14,16)(17,20)(18,24,27,26)(19,28)(21,35,36,32)(22,33)(23,37,40,39)(25,42)(29,45,34,44)(30,31)(38,48)(41,43)(46,47);(1,2)(3,7,10,14)(4,11)(5,6)(8,13)(9,15,12,16)(17,21)(18,25)(19,29)(20,32,33,35)(22,36)(23,38)(24,41,26,42)(27,43)(28,44,31,45)(30,34)(37,47,39,48)(40,46);(1,5,6,2)(3,7,10,14)(4,8)(9,15)(11,13)(12,16)(17,20)(18,26)(19,28)(21,35,36,32)(22,33)(23,39)(24,27)(25,41,43,42)(29,45,34,44)(30,31)(37,40)(38,47,46,48)

Polycyclic presentation:

2^10|b^2=i,c^2=f,d^2=gh,e^2=gij,b^a=bg,c^a=ch,c^b=cgj,d^a=dgh,d^b=dfhi,d^c=dhj,e^a=egij,e^b=egi,e^c=efgij,e^d=ehi,f^a=fi,f^b=fij,f^d=fi,f^e=fij,g^c=gij,g^d=gj,h^b=hij,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285163, #72310874, #72340301, …
maximal normal subgroup of	#72285163, #72310874, #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, 2, F.9);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);