#72285886

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

3000 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,36)(30,40)(31,43)(32,41)(33,39)(34,38)(35,42)(37,45)(44,48)(46,47);(1,3)(2,6,8,9)(4,13,14,10)(5,11)(7,15)(12,16)(17,19)(18,23,26,27)(20,31)(21,30,35,45)(22,29)(24,36,39,43)(25,40)(28,33)(32,46,34,44)(37,42)(38,48)(41,47);(1,2,5,8)(3,6)(4,12,14,7)(9,11)(10,15)(13,16)(17,20)(18,24)(19,28,29,31)(21,34,35,32)(22,33)(23,36,27,43)(25,41,42,38)(26,39)(30,46)(37,48)(40,47)(44,45);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(19,29)(21,35)(23,27)(25,42)(30,45)(37,40)(44,46)(47,48);(1,4)(2,7,8,12)(3,10,11,13)(5,14)(6,15)(9,16)(17,21,22,35)(18,25)(19,30,29,45)(20,32)(23,37)(24,38,39,41)(26,42)(27,40)(28,44)(31,46)(33,34)(36,47,43,48)

Polycyclic presentation:

2^10|b^2=fg,c^2=fj,e^2=ghij,b^a=bf,c^a=cfj,c^b=cgh,d^b=dfhij,d^c=dfj,e^a=efhij,e^b=egi,e^c=efhij,e^d=ehi,f^b=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gj,g^d=gj,g^e=gj,h^b=hij,h^c=hij,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286044, #72310874, #72310969, …
maximal normal subgroup of	#72286044, #72310874, #72310969, …

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