#72284401

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2963 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:

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

Polycyclic presentation:

2^10|b^2=ij,d^2=g,e^2=ghi,b^a=bij,c^b=cf,d^a=dfi,d^b=dfi,d^c=dh,e^a=efhij,e^b=eghij,e^c=ehj,e^d=ehij,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gj,g^e=gj,h^b=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190, #72285259, #72285311, …
maximal normal subgroup of	#72285190, #72285259, #72285311, …

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