Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2709 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15916 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,14)(3,15,18,27,19,28,20,9)(4,10,21,30,17,26,23,31)(5,22,24,29,16,32,25,11);(1,3,6,20,8,19,7,18)(2,9,12,28,13,27,14,15)(4,16,21,24,17,5,23,25)(10,29,30,22,26,11,31,32);(1,4)(2,10)(3,16)(5,19)(6,21)(7,23)(8,17)(9,11)(12,30)(13,26)(14,31)(15,22)(18,25)(20,24)(27,29)(28,32);(1,5)(2,11,13,29)(3,17)(4,19)(6,25)(7,24)(8,16)(9,26,27,10)(12,32,14,22)(15,30,28,31)(18,21)(20,23)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319908, #72319988, #72320075, … |
| maximal normal subgroup of | #72319908, #72319988, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355918);