#72340242

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

2794 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)(8,15)(14,16)(17,18)(19,24)(20,30)(21,33)(22,37)(23,27)(25,36)(26,40)(28,31)(29,32)(34,41)(35,47)(38,43)(39,42)(44,46)(45,48);(1,2,5,7)(3,9)(4,8,12,15)(6,11)(10,16)(13,14)(17,19)(18,24)(20,27)(21,28,36,41)(22,29)(23,30)(25,31,33,34)(26,38)(32,37)(35,44,39,45)(40,43)(42,46,47,48);(1,3)(2,6)(4,13,12,10)(5,11)(7,9)(8,16,15,14)(17,20,23,24)(18,19)(21,34,36,31)(22,38,40,32)(25,41)(26,43)(27,30)(28,33)(29,37)(35,48,39,46)(42,45)(44,47);(2,7)(6,9)(17,21)(18,25)(19,28)(20,31)(22,39)(23,36)(24,34)(26,42)(27,33)(29,45)(30,41)(32,46)(35,40)(37,47)(38,48)(43,44);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,16)(11,13)(17,22,23,40)(18,26,27,37)(19,29,30,43)(20,32,24,38)(21,35,36,39)(25,42,33,47)(28,44,41,45)(31,46,34,48)

Polycyclic presentation:

2^10|b^2=hj,c^2=gij,e^2=ij,b^a=bh,c^a=cf,c^b=cfg,d^a=dij,d^b=dgi,d^c=dij,e^a=ehi,e^c=ehij,e^d=efgij,f^b=fi,f^c=fij,f^e=fj,g^a=gij,g^b=gij,g^e=gj,h^c=hj,h^d=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340248, #72340284, #72340305, …
maximal normal subgroup of	#72340248, #72340284, #72340305, …

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