Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125287 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#469 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5)(4,6)(7,8)(9,10,13,19,15,22,32,18,16,20,31,38,14,21,12,30)(11,23,26,35,28,39,25,36,29,33,40,17,27,34,24,37);(2,5)(3,8)(6,7)(9,11)(10,17)(12,26)(13,24)(14,28)(15,27)(16,29)(18,34)(19,23)(20,35)(21,37)(22,36)(25,31)(30,39)(32,40)(33,38);(2,6)(5,7)(9,12,15,13,16,32,14,31)(10,18,22,38,20,30,21,19)(11,24,28,26,29,25,27,40)(17,33,37,34,35,23,36,39)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32529);