Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,7,8)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32);(1,2,4,3)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,2,4,3)(10,13)(14,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,3,4,2)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=e,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270728);