Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C82 | C23 |
| commutator, socle | C22 | C2×C82 |
| prime radical | C23 | C82 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
248 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(18,22)(23,28)(24,29)(30,32);(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(2,6)(7,12)(8,13)(14,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bi,c^a=ch224 conjugacy classes — view the full character table.
| maximal subgroup of | M322, #72226969, #72226970, … |
| maximal normal subgroup of | M322, #72226969, #72226970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29403);