#72303387

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15986
commutator, Frattini, cosocle	C₂⁶	C₂⁴
prime radical	#62817	C₂²
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

3097 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)(3,7)(4,8)(5,9)(6,10)(11,20)(12,18)(13,19)(14,21)(15,23)(16,22)(17,24)(25,30)(26,31)(27,29)(28,32);(2,7,9,20)(3,11)(8,19,22,31)(10,18,24,30)(12,25)(13,26)(21,32,23,29)(27,28);(3,12)(4,14)(7,18)(8,21)(11,25)(13,26)(15,16)(19,31)(20,30)(22,23)(27,28)(29,32);(2,8,10,23)(4,15)(7,19,18,29)(9,22,24,21)(13,27)(14,16)(20,31,30,32)(26,28)

Polycyclic presentation:

2^10|b^2=g,d^2=h,b^a=be,c^b=ch,d^a=df,d^c=dgh,e^b=ei,e^c=ej,f^c=fij,f^d=fj,g^a=gi,h^a=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303408, #72303433, #72303450, …
maximal normal subgroup of	#72303408, #72303433, #72303450, …

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