Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125419 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1514 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,4,8,5,6,3,7)(9,10)(11,21,23,38)(12,26,28,33)(13,19)(14,17)(15,20)(16,22)(18,29,35,30)(24,34,25,37)(27,36)(31,40)(32,39);(2,6)(3,4)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,31)(18,34)(19,36)(20,32)(21,26)(22,30)(28,40)(29,39)(33,38)(35,37);(1,3,5,4)(2,7,6,8)(10,17,20,19)(12,28)(16,31,32,36)(18,26,35,33)(21,37,38,34)(22,39)(27,40)(29,30)Polycyclic presentation:
2^9|a^2=e,c^2=fg,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32615);