Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10326 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1434 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,5)(4,6)(7,8)(9,10,14,18,15,19,13,28)(11,20,24,31,23,36,22,16)(12,17,26,34,27,33,29,32)(21,35,37,40,25,39,38,30);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,15,23,14,24)(10,16,18,20,19,31,28,36)(12,25,26,38,27,21,29,37)(17,30,32,39,33,40,34,35);(5,8)(9,12)(10,17)(11,21)(13,29)(14,26)(15,27)(16,30)(18,34)(19,33)(20,35)(22,38)(23,25)(24,37)(28,32)(31,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 24816);