#72284616

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

3464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=i,c^2=f,d^2=gh,e^2=j,b^a=bg,c^a=ch,c^b=cgj,d^a=dgh,d^b=dfhi,d^c=dhj,e^a=ehi,e^b=efhi,e^c=efij,e^d=ehi,f^a=fi,f^b=fij,f^d=fi,g^c=gij,g^d=gj,g^e=gj,h^b=hij,h^c=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285163, #72341091, #72341105, …
maximal normal subgroup of	#72285163, #72341091, #72341105, …

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