Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C16 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42×C8 | C22 |
| commutator | C2 | C2×C8×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(2,5)(6,8)(9,10,11,13,12,14,15,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=d,e^2=g,f^2=h,g^2=i,b^a=bd320 conjugacy classes — view the full character table.
| maximal subgroup of | #72210714, #72234228, #72234282 |
| maximal normal subgroup of | #72210714, #72234228, #72234282 |
| maximal quotient of | #72210645, #72234227, #72234237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35399);