Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C42 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#322 |
| prime radical | C23 | #322 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
220 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(1,2,3,8,4,9,11,16,5,7,10,15,12,6,13,14);(2,6,7,9)(3,10)(8,16,15,14)(11,13)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(2,7)(6,9)(8,15)(14,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=eh,d^b=dh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72227018, #72231875, #72231879, … |
| maximal normal subgroup of | #72227018, #72231875, #72231879, … |
| maximal quotient of | #72286315 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29409);