Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102749 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #392 |
| commutator, Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
407 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9)(3,10,13,11)(4,15,6,8)(7,14,16,12)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,32)(29,31)(33,34,37,40)(35,41,42,48)(36,44,46,39)(38,45,47,43);(1,3,5,13)(2,7,9,16)(4,14,6,12)(8,11,15,10)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25)(38,47)(41,48);(1,2)(3,11)(4,8)(5,9)(6,15)(7,12)(10,13)(14,16)(20,30)(22,31)(23,28)(26,29)(33,35,37,42)(34,38,40,47)(36,45,46,43)(39,48,44,41)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,c^b=cf,d^b=dg,f^b=fghi,d^c=dh,e^c=ei,f^c=fghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9975);