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 | C4 | #1368 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7156 |
| prime radical | C23 | #365 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
127 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,14,16,40,19,42,20,51)(4,21,5,25)(6,28)(7,30)(8,13)(9,15,32,44,34,17,35,46)(10,23,37,26)(11,29)(12,31)(18,49,47,57,43,64,50,41)(22,53)(24,56,27,52)(33,62,61,48,39,63,60,45)(36,54)(38,59,58,55);(1,3,5,17,22,43,23,45,7,20,27,46,54,47,55,62,8,19,4,15,6,18,26,48,31,16,24,44,29,50,59,63)(2,9,10,14,36,39,25,57,12,35,38,40,53,61,56,64,13,34,37,42,11,33,21,41,30,32,58,51,28,60,52,49)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fi,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 | #72223653, #72223669, #72223681, … |
| maximal normal subgroup of | #72223653, #72223669, #72223681, … |
| maximal quotient of | #72223652, C2×#94031 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1227);