#72340849

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#37650
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#647907
prime radical	#394108	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5905 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)(3,7)(4,15,17,30)(5,10)(6,9)(8,18,24,31)(11,25,27,29)(12,20)(13,22)(14,21)(16,23)(19,26,32,28);(1,3)(2,7)(4,11)(5,12)(6,14)(8,19)(9,21)(10,20)(13,16)(15,29)(17,27)(18,28)(22,23)(24,32)(25,30)(26,31);(1,4,16,31,6,17,5,18)(2,8,23,30,9,24,10,15)(3,11,13,26,14,27,12,28)(7,19,22,25,21,32,20,29);(2,9)(4,17)(10,23)(18,31);(3,12,14,13)(4,17)(7,20,21,22)(8,24)(11,26,27,28)(15,30)(18,31)(19,25,32,29)

Polycyclic presentation:

2^10|a^2=f,c^2=hi,e^2=ij,h^2=j,b^a=bf,c^a=cfhj,d^a=di,d^b=dg,d^c=di,e^b=eh,e^c=ej,f^c=fj,g^a=gj,g^c=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340854
maximal normal subgroup of	#72340854

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