#72237963

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	#2765029
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#23996	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4511 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=hij,b^a=bfh,c^a=chi,c^b=cg,d^a=dgh,d^b=dgj,d^c=dhij,e^a=eh,e^b=eh,e^c=ehi,f^c=fj,f^d=fj,g^a=gj,g^d=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237964, #72251165, #72251302, …
maximal normal subgroup of	#72237964, #72251165, #72251302, …

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