Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #104373 |
| 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 | |
367 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,6,9)(3,10,12,16)(4,8,5,13)(7,14,15,11)(17,18,21,24)(19,22,26,31)(20,27,28,23)(25,32,29,30)(33,34)(35,39)(36,44)(37,46)(38,42)(40,45)(41,47)(43,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(19,25)(20,28)(22,30)(23,27)(26,29)(31,32)(33,35,38,36)(34,39,42,44)(37,45,47,43)(40,41,48,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,10,15,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30)(33,36,38,35)(34,40,42,48)(37,43,47,45)(39,46,44,41)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,d^c=dh,e^c=ei,f^c=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 11628);