#72237957

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	C₂	#4049127
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=h,c^2=hij,d^2=hi,e^2=hj,b^a=bf,c^a=chi,c^b=cij,d^a=dhij,d^b=dg,d^c=dhi,e^a=ej,e^b=ej,e^c=egh,e^d=ei,f^c=fj,f^d=fj,g^c=gj,g^d=gj,g^e=gj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237964, #72251081, #72251093, …
maximal normal subgroup of	#72237964, #72251081, #72251093, …

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