#72256012

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#405776
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂×#3484	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3949 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)(3,13,16,8)(4,17,20,9)(5,12)(6,10)(7,11)(14,23)(15,27,21,24)(18,32)(19,26,22,28)(25,30)(29,31);(1,3)(2,8)(4,18)(5,15,6,21)(7,16)(9,25)(10,24,12,27)(11,13)(14,19,31,22)(17,32)(20,30)(23,28,29,26);(1,4)(2,9,11,17)(3,14)(5,19)(6,22)(7,20)(8,23,13,29)(10,26,12,28)(15,30)(16,31)(18,21)(24,25,27,32);(1,5,7,6)(2,10,11,12)(3,15,16,21)(4,19,20,22)(8,24,13,27)(9,26,17,28)(14,30,31,18)(23,25,29,32)

Polycyclic presentation:

2^10|a^2=h,b^2=ef,c^2=fhij,d^2=j,g^2=j,b^a=be,c^a=cfj,c^b=cegj,d^a=di,d^b=dhj,e^a=ej,e^c=ei,e^d=ej,f^a=fj,f^c=fj,g^a=gi,g^b=gij,g^c=gi,g^e=gj,h^b=hj,h^c=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256014, #72256062, #72256091, …
maximal normal subgroup of	#72256014, #72256062, #72256091, …

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