Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1893 |
| prime radical | #19128 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 48 points:
(1,2,5,8)(3,6)(4,9,13,14)(7,12)(10,16)(11,15)(17,18,21,26,23,27,22,37)(19,28,31,43,33,44,32,48)(20,34,35,41,29,45,36,25)(24,38,39,46,40,47,42,30);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,29,46,21,31,36,47,23,33,20,30,22,32,35,38)(18,24,25,28,37,42,45,48,27,40,41,44,26,39,34,43);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45745);