Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397260 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14748 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1195 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 1195 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,4,7,12,6,11)(3,13,16,9,18,19,17,14)(5,21,24,10,23,20,22,26)(15,28,29,27,30,25,31,32);(1,3,8,16,7,18,6,17)(2,9,4,19,12,14,11,13)(5,15,24,29,23,30,22,31)(10,25,20,32,26,28,21,27);(1,4,6,12,7,11,8,2)(3,14,16,19,18,9,17,13)(5,20,22,26,23,21,24,10)(15,28,29,32,30,25,31,27);(1,5)(2,10)(3,15)(4,20)(6,22)(7,23)(8,24)(9,25)(11,21)(12,26)(13,27)(14,28)(16,29)(17,31)(18,30)(19,32)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304456);