Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1362 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 238 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:
(1,2,4,10,6,9,16,24)(3,11,14,25,12,21,27,31)(5,7,15,19,17,18,29,30)(8,20,23,26,22,28,32,13);(2,7)(3,12)(4,15)(8,21)(9,18)(11,22)(13,26)(14,28)(16,29)(20,27)(23,32)(25,31);(1,3)(2,8,7,11)(4,14)(5,13)(6,12)(9,22,18,21)(10,23,19,25)(15,20)(16,27)(17,26)(24,32,30,31)(28,29);(2,9)(7,18)(8,22)(10,24)(11,21)(19,30)(23,32)(25,31)Polycyclic presentation:
2^9|a^2=g,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,g^b=gh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97224);