Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,7,3,8)(9,10,12,17,14,18,23,31,13,16,22,28,24,29,35,38)(11,15,20,25,21,27,34,36,19,26,32,30,33,37,40,39);(2,6)(3,4)(7,8)(9,11)(10,15,16,26)(12,20)(13,19)(14,21)(17,30,28,25)(18,27,29,37)(22,32)(23,34)(24,33)(31,39,38,36)(35,40);(2,7)(6,8)(10,16)(11,19)(17,28)(18,29)(20,32)(21,33)(31,38)(34,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31731);