Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,15)(12,13)(14,16)(18,20)(22,24);(2,4)(5,6,7,8)(10,13)(14,15)(17,18,21,20)(19,22,23,24);(1,2)(3,4)(5,6)(7,8)(10,13)(14,15)(19,23)(22,24);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23);(9,11,12,16)(10,14,13,15)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^9|b^2=i,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232034, #72232047, #72232057, … |
| maximal normal subgroup of | #72232034, #72232047, #72232057, … |
| maximal quotient of | #72250140, #72298388, #72323633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 459869);