Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1277 |
| prime radical | C23 | #335 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
138 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,13,15,27)(4,10)(5,20)(6,22)(7,12)(8,14,23,28)(9,18)(11,21)(16,31,29,25)(17,32,30,24)(19,26)(33,34,36,40,37,42,47,54,38,41,48,53,49,55,60,63)(35,43,45,56,46,57,59,64,44,39,50,51,58,52,61,62);(1,3,5,16,21,30,19,28,7,15,18,29,6,17,4,14)(2,8,10,24,22,31,9,13,12,23,26,32,11,25,20,27)(33,35)(34,39,41,43)(36,45)(37,46)(38,44)(40,51,53,56)(42,52,55,57)(47,59)(48,50)(49,58)(54,62,63,64)(60,61)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fh,c^b=ci,d^b=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122);