#72284816

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
prime radical	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2918 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:

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

Polycyclic presentation:

2^10|c^2=gi,e^2=gj,c^a=cf,c^b=cfg,d^b=dfh,d^c=dfhij,e^a=efh,e^b=efh,e^c=eg,e^d=ehj,f^b=fi,f^c=fi,f^e=fij,g^a=gi,g^b=gi,g^d=gj,h^b=hi,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285599, #72340228, #72340301, …
maximal normal subgroup of	#72285599, #72340228, #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, 3, F.7*F.9);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);