Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #331 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#331 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
578 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,13,19,14,18,27,34,15,20,28,36,12,17,25,33)(11,21,23,37,22,32,38,40,24,31,39,35,29,16,26,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,26)(14,29)(15,24)(17,21)(18,31)(19,35)(20,32)(23,25)(27,39)(28,38)(30,33)(34,40)(36,37);(9,12,15,14)(10,17,20,18)(11,22,24,29)(13,25,28,27)(16,21,32,31)(19,33,36,34)(23,38,39,26)(30,37,40,35)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31254);