Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3485 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1134 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,13)(9,22)(10,15,26,27)(11,16)(12,19,29,21)(14,23,33,25)(18,28)(20,31)(24,34)(30,36,32,35);(1,2)(3,4)(5,7,11,8)(6,12,16,29)(9,20,24,18)(10,21,26,19)(13,27,17,15)(14,31,33,28)(22,30,34,32)(23,35,25,36);(2,4)(5,8,11,7)(6,13,16,17)(9,18,24,20)(10,19,26,21)(12,27,29,15)(14,32,33,30)(22,28,34,31)(23,36,25,35);(5,9,11,24)(6,14,16,33)(7,18,8,20)(10,25,26,23)(12,28,29,31)(13,30,17,32)(15,34,27,22)(19,36,21,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408303);