#72285200

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	#245228	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2739 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)(3,5)(4,7)(6,9)(8,15)(10,14)(11,12)(13,16)(20,30)(21,33)(24,31)(25,40)(27,42)(28,44)(32,34)(35,38)(36,47)(39,41)(43,45)(46,48);(1,3)(2,5)(4,11,13,10)(6,8)(7,12,16,14)(9,15)(17,18)(19,23)(20,31,30,24)(21,25,33,40)(22,26)(27,38,42,35)(28,36,44,47)(29,37)(32,41)(34,39)(43,48)(45,46);(1,4)(2,7)(3,10)(5,14)(6,13)(8,11)(9,16)(12,15)(17,19,22,29)(18,23,26,37)(20,27,30,42)(21,28,33,44)(24,38,31,35)(25,36,40,47)(32,45,34,43)(39,46,41,48);(1,5,6,15)(2,8)(3,9)(4,12)(7,11,16,10)(13,14)(17,20,22,30)(18,24)(19,27,29,42)(21,34,33,32)(23,35)(25,41)(26,31)(28,45,44,43)(36,48)(37,38)(39,40)(46,47);(1,5)(2,8,9,3)(4,12,13,14)(6,15)(7,10)(11,16)(17,21)(18,25,26,40)(19,28)(20,32)(22,33)(23,36,37,47)(24,39,31,41)(27,43)(29,44)(30,34)(35,46,38,48)(42,45)

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=gi,e^2=gj,c^b=cf,d^a=dfhj,d^b=dfg,d^c=dfghj,e^a=efj,e^b=efghij,e^c=ef,e^d=eij,f^b=fi,f^d=fij,f^e=fi,g^a=gi,g^b=gij,g^c=gi,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285599, #72340288, #72340301, …
maximal normal subgroup of	#72285599, #72340288, #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, 2, F.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
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);