Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| 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 | |
2387 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 2387 conjugacy classes of subgroups — too many to draw. Only its 282 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:
(9,10,15,20)(11,21,24,30)(12,17,27,34)(13,19,14,18)(16,25,31,23)(22,36,38,35)(26,32,28,33)(29,39,40,37);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,22)(13,25)(14,23)(15,24)(17,29)(18,30)(19,21)(20,31)(26,39)(27,38)(28,37)(32,36)(33,35)(34,40);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,29)(18,32)(19,33)(20,34)(21,35)(23,39)(24,38)(25,37)(30,36)(31,40);(10,18)(13,14)(16,30)(17,32)(19,20)(21,31)(23,25)(26,28)(29,36)(33,34)(35,40)(37,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310546);