#72285289

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

5878 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)(5,15)(6,11)(9,13)(10,14)(12,16)(17,18)(19,24)(20,25)(21,26)(22,34)(23,27)(28,36)(29,37)(30,45)(31,40)(32,41)(33,42)(35,39)(38,46)(43,48)(44,47);(1,3)(2,7)(4,14)(5,12)(6,13)(8,10)(9,11)(15,16)(17,19)(18,24)(20,28,31,35)(21,32)(22,23)(25,36,40,39)(26,41)(27,34)(29,33)(30,44,46,43)(37,42)(38,48,45,47);(1,4)(2,8,11,16)(3,5,13,14)(6,12)(7,10)(9,15)(17,20)(18,25)(19,28,22,35)(21,30)(23,31)(24,36,34,39)(26,38)(27,40)(29,44,32,43)(33,46)(37,48,41,47)(42,45);(1,3)(2,9)(4,14)(5,12)(6,13)(7,11)(8,15)(10,16)(17,21)(18,26)(19,29)(20,30,31,46)(22,32)(23,33)(24,37)(25,38,40,45)(27,42)(28,43,35,44)(34,41)(36,47,39,48);(1,5,6,14)(2,10,11,15)(3,12,13,4)(7,8,9,16)(17,22,23,19)(18,24,27,34)(20,28,31,35)(21,29,33,32)(25,39,40,36)(26,41,42,37)(30,44,46,43)(38,47,45,48)

Polycyclic presentation:

2^10|b^2=f,c^2=fghj,d^2=f,e^2=ij,c^a=ch,c^b=cg,d^a=dfj,d^b=di,d^c=dfgi,e^a=eij,e^b=efi,e^c=efhj,e^d=ei,f^c=fi,g^a=gj,g^b=gi,g^c=gij,g^d=gi,g^e=gij,h^b=hj,h^c=hj,h^d=hij

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