Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| 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 | C23 | #465 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,5,7,6,10,4,9)(3,11,13,8,14,15,12,16)(17,18,21,26,23,27,22,37)(19,28,31,44,29,42,32,47)(20,33,34,39,36,48,35,25)(24,38,41,45,40,46,43,30);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,20,30,21,31,34,38,23,29,36,45,22,32,35,46)(18,24,39,42,37,43,33,44,27,40,25,28,26,41,48,47);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(19,29)(24,40)(28,42)(30,45)(31,32)(38,46)(41,43)(44,47)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=h,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45792);