Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
322 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,6,8)(9,10)(11,21,24,37)(12,18)(13,29)(14,31)(15,20)(16,23,33,38)(17,28)(19,30)(22,34,26,40)(25,32,39,35)(27,36);(2,3)(7,8)(9,11,13,25,30,22,27,38,15,24,28,39,14,26,12,23)(10,16,18,34,31,32,17,21,20,33,36,40,19,35,29,37);(1,3,5,8,4,2,6,7)(11,22,24,26)(12,27)(13,28)(16,32,33,35)(17,29)(18,36)(21,34,37,40)(23,39,38,25)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=cgh,e^b=ei,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35917);