Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19101 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,21,33)(12,24,26,39)(13,27,25,17)(15,28,29,22)(16,31,32,20)(23,38,37,35)(30,36,34,40);(9,11,20,36,13,23,26,28,14,21,31,40,25,37,12,22)(10,15,24,35,17,30,32,33,18,29,39,38,27,34,16,19);(2,5)(3,8)(6,7);(11,21)(15,29)(19,33)(22,28)(23,37)(30,34)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=eg,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309413, #72309422, #72309428, … |
| maximal normal subgroup of | #72309413, #72309422, #72309428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397669);