Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,7)(9,10,12,11)(13,14,17,19,18,22,16,21)(15,23,25,20,26,27,24,28);(1,2)(3,4)(5,7,8,6)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(2,4)(6,7)(13,15,18,26)(14,20,22,28)(16,25,17,24)(19,23,21,27);(9,11,12,10)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=efi,c^2=i,d^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285141);