Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1052 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,21)(11,22,26,38)(12,27,30,19)(13,17,28,18)(14,32,29,20)(16,31,37,23)(24,39,40,36)(25,34,33,35);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,24)(13,31)(14,33)(15,26)(17,34)(18,35)(19,38)(20,39)(21,37)(22,27)(23,28)(25,29)(30,40)(32,36);(10,17)(12,28)(13,30)(14,29)(16,34)(18,21)(19,32)(20,27)(22,39)(23,24)(25,33)(31,40)(35,37)(36,38)Polycyclic presentation:
2^9|a^2=i,c^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=ghi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306951 |
| maximal normal subgroup of | #72306951 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419293);