Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
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 | #334 |
| 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 | |
149 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(2,7,8,9)(3,11)(4,13)(5,12)(10,16,14,15)(17,18,20,24,21,26,31,38,22,25,32,37,33,39,44,47)(19,27,29,40,30,41,43,48,28,23,34,35,42,36,45,46);(1,2,4,9,5,10,13,16,6,8,3,7,12,14,11,15)(17,19)(18,23,25,27)(20,29)(21,30)(22,28)(24,35,37,40)(26,36,39,41)(31,43)(32,34)(33,42)(38,46,47,48)(44,45)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,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, 121);