Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,3,5,4,6,7,8)(9,10)(11,22,25,53)(12,18)(13,29)(14,34)(15,38)(16,21)(17,24,41,59)(19,31)(20,36)(23,51,39,55)(26,62,61,72)(27,63)(28,49)(30,48,60,52)(32,68)(33,50)(35,54,58,45)(37,46)(40,56,43,70)(42,69,71,64)(44,65)(47,57)(66,67);(9,11)(10,17)(12,23)(13,30)(14,35)(15,39)(16,25)(18,40)(19,45)(20,48)(21,41)(22,50)(24,57)(26,36)(27,64)(28,61)(29,53)(31,59)(32,69)(33,60)(34,42)(37,58)(38,43)(44,62)(46,71)(47,54)(49,52)(51,68)(55,63)(56,67)(65,70)(66,72);(9,12,16,15)(10,18,21,38)(11,23,25,39)(13,27,33,32)(14,28,37,36)(17,40,41,43)(19,44,47,66)(20,34,49,46)(22,51,53,55)(24,56,59,70)(26,35,61,58)(29,63,50,68)(30,64,60,69)(31,65,57,67)(42,52,71,48)(45,62,54,72)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dfi,e^b=ei,f^b=fh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232000, #72232003, #72232008, … |
| maximal normal subgroup of | #72232000, #72232003, #72232008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57258);