Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C42 |
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 | |
371 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 371 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:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(20,28)(23,29)(25,32)(27,30)(33,34)(35,41)(36,44)(37,40)(38,42)(39,45)(43,48)(46,47);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,21,24)(19,25,26,32)(20,29,28,23)(22,30,31,27)(36,45)(38,47)(39,44)(42,46);(4,14)(7,12)(9,16)(10,13)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,29)(27,28)(33,35,37,43)(34,38,40,47)(36,46,45,42)(39,48,44,41)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,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, 9901);