Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2656 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4170 conjugacy classes of subgroups; only the 802 normal subgroups are shown.
This group has 4170 conjugacy classes of subgroups — too many to draw. Only its 802 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)(6,8)(10,11)(15,21)(16,23)(19,24)(20,25);(9,10,12,11)(14,18)(19,24)(20,25)(26,28);(5,6,7,8)(9,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(1,2)(3,4)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^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, 4215255);