#72340285

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

2787 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)(10,16)(17,18,23,27)(19,28,32,24)(20,33,25,21)(22,37,38,26)(29,35,40,30)(31,39,46,43)(34,42,41,36)(44,47,48,45);(1,2)(3,9,11,6)(4,13)(5,8)(7,12)(10,14,16,15)(17,19)(18,24,27,28)(20,29,25,40)(21,35)(22,31)(23,32)(26,39,37,43)(30,33)(34,44,41,48)(36,47)(38,46)(42,45);(1,2)(3,6)(4,13,12,7)(5,8)(9,11)(10,14,16,15)(17,20)(18,21)(19,29,32,40)(22,34)(23,25)(24,30,28,35)(26,42)(27,33)(31,44,46,48)(36,37)(38,41)(39,47,43,45);(1,3)(2,6)(4,10,12,16)(5,11)(7,14,13,15)(8,9)(17,21)(18,25)(19,30,32,35)(20,27)(22,36)(23,33)(24,29,28,40)(26,34)(31,45,46,47)(37,41)(38,42)(39,48,43,44);(1,4,5,12)(2,7,8,13)(3,10,11,16)(6,14,9,15)(17,22,23,38)(18,26,27,37)(19,31,32,46)(20,34,25,41)(21,36,33,42)(24,39,28,43)(29,44,40,48)(30,45,35,47)

Polycyclic presentation:

2^10|a^2=i,b^2=f,c^2=g,d^2=g,e^2=ij,b^a=bg,c^a=cgi,c^b=cfg,d^a=di,d^b=dh,d^c=dgi,e^a=ehij,e^b=ej,e^c=efh,e^d=efj,f^a=fi,f^c=fi,f^d=fj,g^b=gi,g^e=gj,h^b=hj,h^c=hij,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340288, #72340307, #72340311, …
maximal normal subgroup of	#72340288, #72340307, #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, 1, F.9);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
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.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 1, 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, 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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);