Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #411625 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1065 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1065 conjugacy classes of subgroups — too many to draw. Only its 206 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:
(12,22)(13,24)(15,25)(16,28)(18,30)(19,31)(21,33)(23,34)(26,37)(29,39)(32,40)(35,38);(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,16)(13,18)(14,17)(19,26)(20,25)(21,27)(22,28)(23,29)(24,30)(31,38)(32,35)(33,36)(34,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,25)(18,27)(22,31)(23,32)(24,33)(28,38)(29,35)(30,36)(34,40)(37,39);(1,4,5,3)(2,6,7,8)(9,12,13,22,14,23,24,34)(10,16,18,28,17,29,30,39)(11,19,21,31,20,32,33,40)(15,26,27,38,25,35,36,37)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gh,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321779);