Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125297 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#329 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,13,18,15,19,30,39,16,21,31,38,14,20,12,28)(11,22,24,40,26,36,23,33,27,35,29,37,25,17,32,34);(2,5)(3,8)(6,7)(9,11)(10,17)(12,29)(13,23)(14,26)(15,25)(16,27)(18,37)(19,35)(20,22)(21,36)(24,30)(28,34)(31,32)(33,39)(38,40);(2,6)(5,7)(9,12,15,13,16,30,14,31)(10,18,20,28,21,38,19,39)(11,23,26,29,27,32,25,24)(17,33,35,40,36,34,22,37)Polycyclic presentation:
2^9|a^2=e,c^2=fgi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,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, 32495);