#72251789

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2109394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#394102	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,c^2=ghij,d^2=ij,h^2=j,b^a=bghj,c^a=cf,c^b=cgi,d^a=dgij,d^b=dhj,d^c=dhi,e^b=egj,e^c=egi,f^a=fj,f^b=fj,f^c=fj,f^d=fj,g^c=gj,h^a=hj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251804, #72251866, #72251931
maximal normal subgroup of	#72251804, #72251866, #72251931

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