Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×C4⋊C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 52 points:
(2,4)(5,6)(7,13)(8,16)(9,11)(10,14)(12,17)(15,19)(18,20)(21,22,24,28,25,29,34,40,26,30,35,41,36,42,46,50)(23,27,32,38,33,39,45,49,31,37,43,47,44,48,51,52);(5,7,8,14,9,15,17,20)(6,10,12,13,11,18,16,19)(21,23)(22,27)(24,32)(25,33)(26,31)(28,38)(29,39)(30,37)(34,45)(35,43)(36,44)(40,49)(41,47)(42,48)(46,51)(50,52);(1,2,3,4)(6,11)(10,18)(12,16)(13,19)(23,31)(27,37)(32,43)(33,44)(38,47)(39,48)(45,51)(49,52)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=eg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42927);