Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10326 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #60741 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1120 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,7,8,5)(4,6)(9,10,14,19,15,20,13,18)(11,21,25,16,24,32,23,31)(12,17,27,34,28,35,29,36)(22,33,26,40,38,37,39,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,33)(18,32)(19,21)(20,31)(22,27)(28,39)(29,38)(30,34)(35,37)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,36)(19,34)(20,35)(21,37)(23,39)(24,38)(25,26)(31,40)(32,33)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18651);