Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1533 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C24 | #100 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 66 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)(9,10,14,18,16,20,29,36,15,21,28,35,13,19,12,27)(11,17,24,30,26,32,39,34,25,33,38,40,23,31,22,37);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,17,21,33)(12,24,29,38)(13,26,16,23)(14,22,28,39)(18,34,35,37)(19,32,20,31)(27,40,36,30);(2,6)(5,7)(9,12,13,28,15,29,16,14)(10,18,20,36,21,35,19,27)(11,22,23,38,25,39,26,24)(17,30,32,34,33,40,31,37)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=gh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32805);