Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35659 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2045 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2045 conjugacy classes of subgroups — too many to draw. Only its 210 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,16)(13,25)(14,18)(15,21)(17,26)(20,36)(22,33)(23,34)(24,32)(27,29)(28,31)(30,37)(35,38)(39,40);(1,3,4,8)(2,5,6,7)(13,26)(15,29)(17,25)(21,27)(23,28)(30,40)(31,34)(37,39);(9,11)(10,15)(12,20)(13,27)(14,22)(16,30)(17,33)(18,29)(19,25)(21,26)(23,39)(24,38)(28,37)(31,35)(32,40)(34,36);(3,8)(5,7)(9,12,14,24)(10,16,18,32)(11,20,22,38)(13,28,26,23)(15,30,29,40)(17,34,25,31)(19,35,33,36)(21,39,27,37)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325085);