#72233853

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	#14911
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	#6453841	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

3340 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,e^2=ij,f^2=i,b^a=bgij,c^a=ce,c^b=cefi,d^a=dgh,d^b=dij,e^b=ei,e^c=eij,e^d=ei,f^a=fj,f^b=fj,f^c=fi,f^d=fi,g^c=gij,h^c=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233858
maximal normal subgroup of	#72233858

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