#72234757

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23900
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4122 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)(4,18)(9,24)(10,27)(11,12)(13,21)(14,31)(19,23)(20,32)(22,28)(25,29)(26,30);(1,2,8,7)(3,13,16,21)(4,10,23,22)(5,12,6,11)(9,15,24,17)(14,26,20,29)(18,28,19,27)(25,32,30,31);(2,7)(3,14,15,31)(4,19)(9,25,21,26)(10,28)(11,12)(13,29,24,30)(16,20,17,32)(18,23)(22,27);(1,3)(2,9)(4,20)(5,16)(6,17)(7,21)(8,15)(10,29)(11,13)(12,24)(14,18)(19,31)(22,30)(23,32)(25,27)(26,28)

Polycyclic presentation:

2^10|b^2=hj,c^2=f,e^2=i,b^a=bhj,c^a=cf,c^b=chj,d^a=dg,d^b=dg,d^c=de,e^a=eh,e^b=ehi,e^c=ej,e^d=ei,f^d=fij,g^b=gj,g^c=ghi,g^e=gj,h^d=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234762, #72234784, #72234789, …
maximal normal subgroup of	#72234762, #72234784, #72234789, …
maximal quotient of	#72237735

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