Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124850 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | #1467 |
| prime radical | C23 | #407 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
269 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(2,5)(3,7)(6,8)(9,10,13,19,14,18,28,34,15,20,27,35,12,16,26,29)(11,21,24,36,25,17,30,32,22,31,37,39,23,33,38,40);(1,2,4,6)(3,8,7,5)(10,16,20,18)(11,22)(13,27)(17,31,33,21)(19,29,35,34)(23,25)(26,28)(32,39,40,36);(9,11,14,25,15,22,12,23)(10,17,16,21,20,33,18,31)(13,24,28,30,27,37,26,38)(19,32,29,36,35,40,34,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=fgh,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=cfgh,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.
| maximal subgroup of | #72239985, #72239989, #72239990, … |
| maximal normal subgroup of | #72239985, #72239989, #72239990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32254);