Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C23 | #443 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,11,13,24,15,25,14,20)(12,23,26,34,28,35,27,31);(1,2,3,4)(5,6)(7,12)(8,20)(9,13)(10,15)(11,16)(14,21)(17,31)(18,26)(19,28)(22,24)(23,29)(25,30)(27,32)(33,34)(35,36);(5,7,16,29,9,18,22,33,10,19,30,36,21,32,8,17)(6,12,11,23,13,26,24,34,15,28,25,35,14,27,20,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,g^2=i,b^a=bd,d^b=dg,e^b=eg,g^b=gi,h^b=hi176 conjugacy classes — view the full character table.
| maximal subgroup of | #72216211, #72216270, #72216273, … |
| maximal normal subgroup of | #72216211, #72216270, #72216273, … |
| maximal quotient of | #72225365 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36338);