#72285183

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

2827 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)(4,8)(5,9)(6,10)(11,15)(12,14)(13,16)(18,23)(21,33)(24,37)(25,38)(31,36)(39,41)(40,43)(47,48);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,14,9,15)(17,18,22,23)(19,24,29,37)(20,25,32,38)(21,34,33,26)(27,41,30,39)(28,40,35,43)(31,42,36,44)(45,47,46,48);(1,4)(2,8)(3,11,13,12)(5,6)(7,14,16,15)(9,10)(17,19)(18,24)(20,30)(21,35,33,28)(22,29)(23,37)(25,41)(26,43,34,40)(27,32)(31,46,36,45)(38,39)(42,48,44,47);(1,4,6,5)(2,8,10,9)(3,11)(7,15)(12,13)(14,16)(17,20,22,32)(18,25,23,38)(19,27,29,30)(21,36)(24,39,37,41)(26,44)(28,45)(31,33)(34,42)(35,46)(40,47)(43,48);(1,5,6,4)(2,9)(3,12,13,11)(7,15)(8,10)(14,16)(17,21)(18,26)(19,28,29,35)(20,31,32,36)(22,33)(23,34)(24,40,37,43)(25,42,38,44)(27,45)(30,46)(39,47)(41,48)

Polycyclic presentation:

2^10|b^2=ij,c^2=g,d^2=gij,e^2=ghi,b^a=bj,c^a=cg,c^b=cf,d^b=dfgh,d^c=dij,e^a=efhij,e^b=eghij,e^c=ehij,e^d=ehj,f^a=fi,f^c=fi,f^d=fij,f^e=fij,g^b=gi,g^e=gj,h^a=hi,h^b=hi,h^c=hj,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526, #72285553, #72285603, …
maximal normal subgroup of	#72285526, #72285553, #72285603, …

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