Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
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 | |
341 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,6,7,8)(9,10,13,19,14,18,29,34,15,17,28,33,12,25,26,35)(11,20,23,36,24,32,38,40,21,37,27,39,22,16,30,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16,17,32)(12,22)(13,27)(14,24)(15,21)(18,20,25,37)(19,31,33,40)(23,28)(26,38)(29,30)(34,36,35,39);(1,3,5,4)(2,6,7,8)(10,17)(11,21)(13,28)(18,25)(22,24)(26,29)(31,40)(36,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efgi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31561);