Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C2×D16 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
171 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7)(3,6,12,14)(5,13,11,9)(8,16,15,10)(17,18,20,24,21,25,28,31,22,19,23,26,29,27,30,32);(2,6)(3,10)(5,11)(7,14)(8,9)(12,16)(13,15);(2,7)(6,14)(8,15)(9,13)(17,19,20,26,21,27,28,32,22,18,23,24,29,25,30,31)Polycyclic presentation:
2^9|a^2=e,c^2=eh,d^2=fh,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fh,d^b=df,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#122946, #72225391, #72226920, … |
| maximal normal subgroup of | C2×#122946, #72225391, #72226920, … |
| maximal quotient of | #72225369, C2×#122946, #72230700 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30142);