#72306624

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#499831
commutator, Frattini, cosocle	#532	C₂⁴
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3444 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=ij,d^2=h,g^2=i,h^2=j,b^a=bj,c^a=ce,c^b=cf,d^a=di,d^b=dg,e^b=ei,e^d=ej,f^a=fij,f^b=fj,f^d=fh,f^e=fj,g^b=gi,g^c=ghij,g^f=gi,h^b=hi,h^f=hj,i^c=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306628, #72306636, #72306739, …
maximal normal subgroup of	#72306628, #72306636, #72306739, …

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