#72284797

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

3343 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)(3,4)(5,13)(6,10)(7,8)(9,15)(11,16)(12,14)(18,23)(19,20)(26,29)(31,32);(1,3)(2,7)(4,10)(5,14,15,11)(6,8)(9,12,13,16)(17,18)(19,25)(20,24)(21,29,30,26)(22,23)(27,31,28,32);(1,4,6,7)(2,8,10,3)(5,12)(9,11)(13,14)(15,16)(17,19,22,20)(18,24,23,25)(21,28)(26,31)(27,30)(29,32);(1,5,6,15)(2,9)(3,11,8,14)(4,12)(7,16)(10,13)(17,20,22,19)(18,25)(21,27)(23,24)(26,31,29,32)(28,30);(2,10)(5,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)

Polycyclic presentation:

2^10|b^2=f,c^2=fij,d^2=gij,b^a=bij,c^a=cfij,c^b=cfij,d^a=dfj,d^b=dh,d^c=dfg,e^a=ehi,e^b=efg,e^c=efgh,e^d=ehi,f^d=fi,f^e=fj,g^a=gi,g^b=gj,g^c=gi,g^e=gj,h^b=hi,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285190, #72310982, #72338825, …
maximal normal subgroup of	#72285190, #72310982, #72338825, …

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