#72282461

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1898
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	#6470077	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(2,5)(3,6)(7,8)(9,10)(11,19)(12,17,13,16)(14,18)(15,22,24,23)(20,21);(9,10)(11,20)(12,17,13,16)(14,18)(15,23,24,22)(19,21);(1,2,6,8,4,7,3,5)(9,11)(10,15)(12,22)(13,23)(14,21)(16,19)(17,20)(18,24);(11,21)(15,24)(19,20)(22,23)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=efij,e^2=i,c^a=ce,c^b=cf,d^c=dj,e^a=eh,e^b=ehi,f^a=fhi,f^b=fhi,g^c=ghi,g^e=gj,g^f=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282500, #72282619, #72282838
maximal normal subgroup of	#72282500, #72282619, #72282838

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