Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #410818 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15091 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1251 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 1251 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:
(2,8)(3,12)(5,6)(9,13)(10,24)(11,20)(14,27)(17,19)(21,25)(22,31)(23,26)(28,32);(1,2,6,20,7,11,5,8)(3,13,16,9,12,26,15,23)(4,10,17,24,18,25,19,21)(14,28,29,22,27,31,30,32);(1,3,6,16,7,12,5,15)(2,9,8,13,11,23,20,26)(4,14,17,29,18,27,19,30)(10,22,21,28,25,32,24,31);(1,4)(2,10)(3,14)(5,19)(6,17)(7,18)(8,21)(9,22)(11,25)(12,27)(13,28)(15,30)(16,29)(20,24)(23,32)(26,31)Polycyclic presentation:
2^9|b^2=gh,c^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318014);