Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#547 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,7,9,10,12,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(2,4)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27);(6,10)(7,11)(15,23)(18,21)(22,28)(25,27);(5,7,8,10,9,11,12,6)(14,20)(18,21)(19,26)(25,27)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,d^b=df,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306387);