Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511954 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2215 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,20)(15,22)(16,21)(17,26)(18,24)(23,28)(25,27);(5,7,8,11)(6,9,10,12)(13,14,21,25,18,22,26,28,19,23,17,15,24,27,16,20);(6,10)(9,12);(2,4)(13,15,16,23,24,28,17,22,19,25,26,14,18,20,21,27)Polycyclic presentation:
2^9|b^2=efh,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72319432, #72319567, #72319574, … |
| maximal normal subgroup of | #72319432, #72319567, #72319574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419519);