Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C22×C4 | C2×C16 |
| Frattini, cosocle | C2×C22⋊C16 | C22 |
| socle | C22 | #1314 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C22⋊C16 ◁ G | |
| LECS | G ▷ C2×C22⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,14,20,32,45,15,21,33,46,35,48,60,68,16,22,34,47,36,49,61,69,12,18,29,42,30,43,57,66)(11,23,25,50,26,51,54,70,24,40,52,44,53,64,71,67,27,41,31,58,55,65,59,72,28,17,37,38,56,39,62,63);(2,5)(6,7)(9,11)(10,17)(12,28)(13,31)(14,26)(15,24)(16,27)(18,41)(19,44)(20,39)(21,23)(22,40)(25,34)(29,52)(30,56)(32,59)(33,37)(35,53)(36,55)(38,47)(42,50)(43,65)(45,67)(46,58)(48,51)(49,64)(54,61)(57,71)(60,62)(63,69)(66,70)(68,72)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=ci,d^b=dei,d^c=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 866);