#72312749

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15257
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

4334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=h,c^2=ij,f^2=j,b^a=bh,c^a=cf,d^b=dg,d^c=de,e^a=ei,e^b=eij,f^a=fj,f^c=fj,f^d=fi,g^a=gi,g^b=gi,g^c=gij,h^d=hi

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312782, #72312857, #72312936
maximal normal subgroup of	#72312782, #72312857, #72312936

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