Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 830 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15,10,16,9,14)(7,17,18,12,21,27,20,28)(8,13,23,30,24,31,25,32)(19,33,34,26,35,29,22,36);(5,7)(6,12)(8,22)(9,18)(10,21)(11,20)(13,29)(14,27)(15,17)(16,28)(19,25)(23,35)(24,34)(26,30)(31,33)(32,36);(7,18)(8,23)(9,11)(12,17)(13,30)(14,15)(19,35)(20,21)(24,25)(27,28)(29,33)(31,32);(2,4)(5,8,10,24)(6,13)(7,19,21,35)(9,25,11,23)(12,26)(14,32)(15,30)(16,31)(17,33)(18,34,20,22)(27,29)(28,36)Polycyclic presentation:
2^9|a^2=fg,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386426);