Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| 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 | |
226 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 226 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:
(1,2,4,10,5,11,20,33,6,12,21,34,22,35,49,58,7,8,19,26,23,27,42,52,24,28,48,53,50,36,59,61)(3,13,15,38,16,39,43,62,17,25,44,32,45,51,63,57,14,37,18,46,40,60,47,64,41,9,29,30,54,31,55,56);(2,8)(3,14)(11,27)(12,28)(15,18)(16,40)(17,41)(29,44)(30,32)(35,36)(38,46)(43,47)(45,54)(55,63)(56,57)(62,64);(1,3,5,16,6,17,22,45,7,14,23,40,24,41,50,54)(2,9,27,51,12,13,36,60,8,25,11,31,28,37,35,39)(4,18,20,47,21,29,49,55,19,15,42,43,48,44,59,63)(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,c^b=ci,e^b=ef,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287401, #72342978, #72343286, … |
| maximal normal subgroup of | #72287401, #72342978, #72343286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56928);