#72285769

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

2838 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)(3,7)(4,6)(5,12)(8,13)(9,11)(10,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,36)(23,27)(28,44)(29,39)(30,38)(31,40)(32,37)(33,41)(34,43)(35,42)(45,47)(46,48);(7,9)(8,12)(10,16)(14,15)(17,19)(18,24)(20,32)(21,30)(22,29)(23,31)(25,37)(26,39)(27,40)(28,35)(33,45)(34,46)(36,38)(41,48)(42,44)(43,47);(1,3)(2,7)(4,9)(5,10,13,16)(6,11)(8,14,12,15)(17,20)(18,25)(19,28,31,32)(21,34)(22,33)(23,35)(24,37,40,44)(26,43)(27,42)(29,46,30,45)(36,41)(38,48,39,47);(1,4,6,2)(3,9,11,7)(5,8)(10,14)(12,13)(15,16)(17,21)(18,26)(19,29,31,30)(20,33)(22,23)(24,38,40,39)(25,41)(27,36)(28,45,32,46)(34,35)(37,47,44,48)(42,43);(1,5)(2,8,4,12)(3,10)(6,13)(7,14,9,15)(11,16)(17,22,23,21)(18,26)(19,30)(20,34)(24,39,40,38)(25,41,42,43)(27,36)(28,45,32,46)(29,31)(33,35)(37,47)(44,48)

Polycyclic presentation:

2^10|c^2=g,d^2=gj,e^2=ghi,b^a=bf,c^a=cg,c^b=ch,d^a=dgj,d^b=df,d^c=di,e^a=ef,e^b=efh,e^c=efgj,e^d=ehij,f^c=fj,f^d=fi,f^e=fij,g^b=gi,g^e=gj,h^a=hij,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340284, #72340293, #72340307, …
maximal normal subgroup of	#72340284, #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, 3, F.7);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*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, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, 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);