#72251928

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#47031
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#4320312
prime radical	#394195	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3757 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:

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

Polycyclic presentation:

2^10|c^2=fg,d^2=fh,e^2=i,h^2=j,c^b=cf,d^a=dfgj,d^b=dfgi,d^c=dghj,e^a=eg,e^b=egij,e^c=ehi,e^d=egij,f^d=fj,f^e=fj,g^d=gj,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251931, #72251940, #72251951, …
maximal normal subgroup of	#72251931, #72251940, #72251951, …

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