Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7457 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1368 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| prime radical | C23 | #365 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
132 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 132 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(6,12,13,14,16,30,17,31)(7,18,20,34)(8,22,23,24)(9,25)(10,26)(15,32,28,36,27,35,33,29)(19,21);(2,4)(5,6,8,14,9,15,24,32,10,17,20,30,25,33,34,36,11,16,23,31,19,27,22,35,26,13,7,12,21,28,18,29)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=f,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fh,h^a=hi,d^b=dh,g^b=gi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216304, #72216317, #72216330, … |
| maximal normal subgroup of | #72216304, #72216317, #72216330, … |
| maximal quotient of | #72223652, C2×#94030, #72242136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1226);