Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #407027 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14945 |
| 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 | |
1188 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 208 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)(3,13)(4,17)(5,12)(6,11)(7,10)(8,15)(9,19)(14,22)(16,27)(18,25)(20,26)(21,23)(24,30)(28,29)(31,32);(1,3,7,16)(2,8,10,23)(4,14)(5,21,6,15)(9,24)(11,27,12,13)(17,28)(18,31)(19,29)(20,30)(22,25)(26,32);(1,4)(2,9)(3,14,16,30)(5,19)(6,18)(7,20)(8,22,23,24)(10,25)(11,17)(12,26)(13,28,27,32)(15,31,21,29);(2,10)(5,6)(8,23)(9,25)(15,21)(18,19)(22,24)(29,31)Polycyclic presentation:
2^9|b^2=efh,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314223);