Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,4,7,5,3,8,6)(9,10)(11,22,25,52)(12,18)(13,29)(14,34)(15,38)(16,21)(17,24,41,57)(19,31)(20,36)(23,50,39,54)(26,60,59,72)(27,61)(28,49)(30,48,58,51)(32,68)(33,65)(35,53,56,45)(37,46)(40,55,43,70)(42,69,71,62)(44,63)(47,67)(64,66);(1,3)(2,5)(4,6)(7,8)(9,11,16,25)(10,17,21,41)(12,23,15,39)(13,30,33,58)(14,35,37,56)(18,40,38,43)(19,45,47,53)(20,48,49,51)(22,29,52,65)(24,31,57,67)(26,28,59,36)(27,62,32,69)(34,42,46,71)(44,60,64,72)(50,61,54,68)(55,63,70,66);(2,3)(6,7)(9,12,16,15)(10,18,21,38)(11,23,25,39)(13,27,33,32)(14,28,37,36)(17,40,41,43)(19,44,47,64)(20,34,49,46)(22,50,52,54)(24,55,57,70)(26,35,59,56)(29,61,65,68)(30,62,58,69)(31,63,67,66)(42,51,71,48)(45,60,53,72)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=dfi,e^b=ei,f^b=fh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72234358, #72234365, #72234376, … |
| maximal normal subgroup of | #72234358, #72234365, #72234376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57355);