Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1944 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1944 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(3,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72282933, #72282959, #72283093, … |
| maximal normal subgroup of | #72282933, #72282959, #72283093, … |
| maximal quotient of | #72284088 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325409);