Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 158 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 — 4 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(18,23)(24,31)(25,30)(26,32);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,18,20,23)(19,25,22,30)(21,24,28,31)(26,27,32,29);(10,14)(13,15)(18,24)(19,22)(21,29)(23,31)(25,26)(27,28)(30,32);(1,4,5,3)(2,6,7,8)(17,19,20,22)(18,25,23,30)(21,27,28,29)(24,32,31,26)Polycyclic presentation:
2^9|a^2=ei,b^2=e,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72343349, #72343363, #72343370, … |
| maximal normal subgroup of | #72343349, #72343363, #72343370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406870);