Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1624 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10432 |
| prime radical | #28249 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 32 points:
(1,2)(3,16)(4,20)(5,25)(6,17)(7,13)(8,12)(9,15)(10,18)(11,23)(14,27)(19,29)(21,22)(24,31)(26,32)(28,30);(1,3)(2,10)(4,15)(5,20)(6,11)(7,28)(8,18)(9,21)(12,22)(13,23)(14,29)(16,25)(17,24)(19,26)(27,31)(30,32);(1,4,5,22,9,10,8,16)(2,11,25,28,15,19,12,31)(3,17,20,13,21,32,18,14)(6,24,7,23,26,30,27,29)Polycyclic presentation:
2^9|c^2=dgh,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,g^a=gi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,e^c=eg,f^c=fh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52355);