#72338217

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7555
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=eg,c^2=defi,d^2=j,e^2=i,f^2=j,b^a=be,c^a=ceh,c^b=cd,d^a=di,d^c=di,e^a=ei,f^b=fij,f^c=fi,g^c=gj,h^b=hi,h^c=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338218, #72338251, #72338258
maximal normal subgroup of	#72338218, #72338251, #72338258

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