Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29,31,40)(20,26)(21,27)(22,28)(23,25)(24,30,37,33)(32,43,45,48)(34,41)(35,42)(36,39)(38,44,47,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,32,22,30,35,44,23,31,36,45,20,33,34,46)(18,24,27,38,28,29,42,43,25,37,39,47,26,40,41,48);(3,9)(6,12)(10,15)(13,16)(18,25)(24,37)(26,28)(27,39)(29,40)(38,47)(41,42)(43,48)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,c^b=cg,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33775);