Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator, prime radical | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
660 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 40 points:
(1,2)(3,4,6,8)(5,7)(9,10,13,18,14,17,28,34)(11,19,22,35,21,15,27,30)(12,23,26,37,25,16,31,33)(20,29,36,39,24,32,38,40);(1,3)(2,4,7,8)(5,6)(9,11)(10,15)(12,24,25,20)(13,27)(14,21)(16,32,23,29)(17,19)(18,35)(22,28)(26,36,31,38)(30,34)(33,39,37,40);(1,4)(2,6,7,3)(5,8)(9,12)(10,16,17,23)(11,20)(13,26)(14,25)(15,29,19,32)(18,33,34,37)(21,24)(22,36)(27,38)(28,31)(30,39,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fh,g^b=gh,e^c=eh,f^c=fi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27831);