Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 805306368 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4926 conjugacy classes of subgroups; only the 1074 normal subgroups are shown.
This group has 4926 conjugacy classes of subgroups — too many to draw. Only its 1074 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(10,11)(13,14,18,21)(15,16,19,24)(17,25,26,20)(22,28,27,23);(5,6,8,7)(13,15)(14,16)(17,22)(18,19)(20,23)(21,24)(25,28)(26,27);(9,10,12,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,25)(26,28);(1,2,3,4)(5,7,8,6)(9,11,12,10);(13,17,18,26)(14,20,21,25)(15,22,19,27)(16,23,24,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215642);