Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500599 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
595 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 595 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(10,13)(11,16)(12,14)(17,18)(19,20)(21,23)(22,24);(1,2,4,6)(3,7,8,5)(9,10,12,13,15,18,14,17)(11,19,21,24,22,23,20,16);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,23)(14,21)(15,22)(17,19)(18,24);(9,10,14,17,15,18,12,13)(11,16,21,19,22,24,20,23)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265494, #72265596, #72265615, … |
| maximal normal subgroup of | #72265494, #72265596, #72265615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414776);