Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | #11488 |
| prime radical | #525 | C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 233 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 — 3 generators on 64 points:
(1,2,4,10,5,11,20,33,6,12,21,34,22,35,49,58,7,8,19,26,23,27,40,52,24,28,48,53,50,37,60,61)(3,13,14,38,15,39,41,62,16,25,42,32,43,51,63,57,17,36,18,46,44,59,47,64,45,9,29,30,54,31,55,56);(2,8)(9,25)(11,27)(12,28)(13,36)(31,51)(35,37)(39,59);(1,3,5,15,6,16,22,43,7,17,23,44,24,45,50,54)(2,9,27,51,12,13,37,59,8,25,11,31,28,36,35,39)(4,18,20,47,21,29,49,55,19,14,40,41,48,42,60,63)(10,32,52,56,34,46,61,62,26,30,33,57,53,38,58,64)Polycyclic presentation:
2^9|a^2=e,c^2=fg,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,e^b=ef,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287376, #72321005, #72321032, … |
| maximal normal subgroup of | #72287376, #72321005, #72321032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56927);