#72285182

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5880 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18)(19,24)(20,25)(21,33)(22,36)(23,27)(26,35)(28,38)(29,44)(30,40)(31,41)(32,42)(34,43)(37,39)(45,47)(46,48);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,13,15,16)(17,19)(18,24)(20,30,32,28)(21,29,35,37)(22,23)(25,40,42,38)(26,39,33,44)(27,36)(31,46)(34,45)(41,48)(43,47);(1,4)(2,7)(3,9,10,11)(5,12)(6,13,14,16)(8,15)(17,20)(18,25)(19,28)(21,34,35,31)(22,30)(23,32)(24,38)(26,43,33,41)(27,42)(29,46,37,45)(36,40)(39,48,44,47);(3,10)(4,12)(7,15)(9,11)(17,21)(18,26)(19,29)(20,31)(22,37)(23,35)(24,39)(25,41)(27,33)(28,45)(30,46)(32,34)(36,44)(38,47)(40,48)(42,43);(4,12)(7,15)(9,11)(13,16)(17,22,23,19)(18,24,27,36)(20,30,32,28)(21,29,35,37)(25,38,42,40)(26,44,33,39)(31,45,34,46)(41,48,43,47)

Polycyclic presentation:

2^10|b^2=f,c^2=h,e^2=i,c^a=ch,c^b=cg,d^a=dfhij,d^b=dgij,d^c=dfh,e^a=ei,e^b=efij,e^c=ej,e^d=ei,f^c=fi,f^d=fi,g^a=gj,g^b=gi,g^c=gj,g^e=gi,h^b=hj,h^d=hi

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185
maximal normal subgroup of	#72286185

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