#72233362

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#62817	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

2961 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,e^2=j,b^a=bg,c^a=cf,c^b=ce,d^a=dij,d^b=dgi,e^a=eh,e^b=ei,e^c=ej,e^d=eij,f^a=fij,f^b=fhi,f^e=fj,g^c=gij,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233377, #72233409, #72233437
maximal normal subgroup of	#72233377, #72233409, #72233437

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