Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102685 |
| 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 | |
351 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,18,22,20)(19,25,28,32)(21,29,31,24)(23,30,26,27)(33,34)(35,38)(36,39)(37,41)(40,45)(42,46)(43,48)(44,47);(1,2)(3,11)(4,10)(5,7)(6,9)(8,12)(13,15)(14,16)(17,19,22,28)(18,23,20,26)(21,30,31,27)(24,32,29,25)(33,35,37,43)(34,38,41,48)(36,44,45,42)(39,47,40,46);(1,3,6,13)(2,8,9,16)(4,12,5,14)(7,11,10,15)(17,20,22,18)(19,26,28,23)(21,24,31,29)(25,30,32,27)(34,39)(36,45)(38,46)(40,41)(42,44)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=g,e^2=i,f^2=gh,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=fgh,d^c=dh,e^c=ei,f^c=fgh41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9900);