#72256398

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23900
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#6499871	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3832 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:

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

Polycyclic presentation:

2^10|a^2=hj,d^2=h,b^a=bf,c^a=ce,c^b=cg,d^a=dj,d^b=dfj,d^c=deghi,e^a=ej,e^b=eh,e^d=ehi,f^c=fij,g^a=ghi,g^d=ghi,g^e=gj,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256410
maximal normal subgroup of	#72256410

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