#72284865

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
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,10)(4,8)(5,9)(6,13)(7,14)(11,15)(12,16)(17,18)(19,23)(20,24)(21,27)(22,25)(26,28)(29,31)(30,32);(1,2)(3,6)(4,7,14,8)(5,9)(10,13)(11,16,12,15)(17,19)(18,23)(20,26,29,32)(21,25)(22,27)(24,30,31,28);(1,3)(2,6)(4,12)(5,13)(7,15)(8,16)(9,10)(11,14)(24,31)(28,30);(1,4,5,14)(2,7)(3,11)(6,15,10,16)(8,9)(12,13)(17,20)(18,24,25,31)(19,26,27,32)(21,30)(22,29)(23,28);(1,4)(2,8,9,7)(3,12,13,11)(5,14)(6,16)(10,15)(17,21,22,23)(18,19)(20,28,29,30)(24,32)(25,27)(26,31)

Polycyclic presentation:

2^10|b^2=f,d^2=gij,e^2=gj,b^a=bf,c^a=cfij,d^a=dfj,d^b=dfg,d^c=dfgh,e^a=efhij,e^b=efghij,e^c=efij,e^d=ei,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gij,h^b=hj,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190, #72285526, #72340481
maximal normal subgroup of	#72285190, #72285526, #72340481

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