Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C22×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,4,7,5,6,8,3)(9,10)(11,13,21,28)(12,22,25,34)(14,17)(15,18)(16,24,33,30)(19,36)(20,27,29,38)(23,37)(26,39)(31,35)(32,40);(1,3,4,6,5,7,8,2)(9,11,13,18,15,21,28,10)(12,19,26,33,25,32,23,16)(14,29,27,17,31,20,38,35)(22,30,37,40,34,24,39,36);(9,12)(10,16)(11,19)(13,26)(14,30)(15,25)(17,34)(18,33)(20,39)(21,32)(22,35)(23,28)(24,31)(27,40)(29,37)(36,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44882);