Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C23 | C2×SD32 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
217 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(2,6,7,8)(9,10,13,19,16,23,39,22)(11,24,26,58,29,61,37,60)(12,30,33,67,36,53,72,68)(14,40,38,20,15,42,32,21)(17,44,45,25,49,62,48,64)(18,50,52,35,55,70,54,34)(27,65,41,59,28,66,43,51)(31,56,69,46,63,57,71,47);(1,2)(3,6)(4,8)(5,7)(9,11,16,29)(10,17,23,49)(12,31,36,63)(13,37,39,26)(14,41,15,43)(18,51,55,59)(19,48,22,45)(20,56,21,57)(24,30,61,53)(25,34,64,35)(27,32,28,38)(33,71,72,69)(40,46,42,47)(44,50,62,70)(52,66,54,65)(58,68,60,67);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fghi,d^2=gh,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57582);