Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1331 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 50 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)(9,10,13,19,15,20,35,48,16,22,36,49,39,21,14,37)(11,23,25,51,26,52,57,71,28,53,58,72,27,59,38,44)(12,29,31,45,33,60,66,47,34,62,67,46,68,61,32,18)(17,30,40,54,42,63,69,56,43,65,70,55,50,64,41,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,30)(13,25)(14,38)(15,26)(16,28)(18,44)(19,40)(20,42)(21,50)(22,43)(23,29)(24,32)(27,39)(31,54)(33,63)(34,65)(35,57)(36,58)(37,41)(45,51)(46,72)(47,71)(48,69)(49,70)(52,60)(53,62)(55,67)(56,66)(59,61)(64,68)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=g,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1113);