Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,5,8,6,3,4,7)(9,10)(11,19,21,36,23,35,22,33)(12,17)(13,18)(14,16)(15,20,29,37,28,38,30,24)(25,32)(26,34)(27,31)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,24)(13,21)(14,23)(16,28)(17,33)(18,29)(19,34)(20,26)(22,25)(27,37)(30,32)(31,36)(35,40)(38,39);(1,4,6,5)(2,7,3,8)(10,16)(15,28)(17,31)(18,32)(19,35)(29,30)(33,36)(34,40)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31300);